رياضي عمومي 1(Calculus I)

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Author: مهدي نجفي خواه

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=nm ≥ nl G d Tm ≥ l < n ≥ S w0) =m < l < n B< l ∈ Q G d Tm < n S w0

= ~ = ~ · · · · · · = ~

· · · · · ·

TV+N .C & %D Q & & &
D!3 < V+& n 8& S C %! d & m T>I~90 9090 S C < U.B fke TV+

!. p & ! `>~0 9600

T < m ≤ |n| < m, n ∈ Z S $ " %: $ r < p & j3 S C G d @ .+/& r < %./ | p = ≤ r < m < n = mp + r =V+ m _& n V+N

"

=

+ −

√ √

=

√ + √ −

+P w

+&+& X β < α S C w0 √ = + = α+β

√

x − x

+

w e jCx wJ

=n × n = G d Tn = S w& c85 B<x w )

V+P +& C ! %./ k5& - < +Q %" " %! +Q ^+.P & =

^

= n=n

_+ < 5+DX C P ./0 - ( ) *+ , - T5x %! ' Y+34 C P ^ S C & Q< qwP 4 [Q( / 5 & =%+ %! p C 2j& S C ,e C & +h+& $ =V+ < mH = + $ =V+& ^&& = = = < V+ < & VP L& T = + = + = + + + = + + = + + = + + + == =

F G & ! 0"

x − = @Q5 S P w> - fk4 d =+&+&

V+ [5 n ∈ N i< 5+DX C @L & w9 n P L & + %&E =Hn = + + · · · + n =%! Y+34 +l S C ' Hn p < < & ! ' p + i: n+ q n q p − " ! L mH +P =%! q n+ =%! 8$ p L W4 " 4

p @! P @L& + %&E @N ! & mK! q

w6

>

%: $ nk , · · · , n

<] %! 5+DX C ' n + i: w %&E =%" . m ∈ N m 8& 5 =%+ √ =%+ S n +

Z @+7

√ Q( )

, n

5+DX C T

p = + + ···+ q n n nk

@C. +P w0 =%! >=>= n<

=

+

<

p < q

"

,e C &

= + + + +

√

+P mK! < + [5 Q( ) .P Q(π) w> =%! >=>= n< Z @+7

S

+

√

+

−

=L&

√

√

V+ Q -" G -%! S C d @;& c < b < f < a < & @<_QVh/ 7e ' a = & b = c = L & < T%! /& a = b + c √ √ " i: S L T%+ S , P a = √ √ T"& " " ! 24 & = n < ∈ Q m √ + < vX < ! < & & =m = n V < %! |
√ √ & Q L C. ^ Q( ) = a + b a, b ∈ Q %! √ √ ^ ,@! mH = ∈/ Q( ) L T%! +5 LP Q< %! √ ' _+ O& @C. & : y- & $z %! %! %/ |fC

C +P w9 =%!

√ Q( + ) @C.

w6

√

^ T%+ S V. P W(& ^ & √ ' +p @;N S C Q @C. %! 5 & = N+N C @C. 8 ^ = B< ="& L+ L [5 & T{7; ^ Y+ &

$B

4 s C =A ⊆ R + i: L & V+P sup A . & < S A =x ≤ s x ∈ A P x ∈ A ' < P L & S A 4 s C =s − < x " %: V+P inf A . & < =s ≤ x x ∈ A P L & " %: x ∈ A ' < P L & =s + > x

23(! F0BC!

24

F

√ √ √ √ √ Q( , ) = a + b + c + d |a, b, c, d ∈ Q}

23(! 24

$ ! mH =" _+ DQB π C g & ! ^ < - & V+ U: L [5 & 8 ^

&H I!

P L & V+hS A ' 4 s C V+hS A ' 4 s C =x ≤ s x ∈ A 4 A @C. =s ≤ x x ∈ A P L &

@C. ="& " O& ' / V+hS ^+hH ' / V+hS 4 A ="& "

>!

CJ I!

!"! J >!

± ak ×

!"! H

2DC 4

k + · · · + a × + a

+b ×

+b ×

+ · · · + bn ×

+ ···

@C. =V+P ±ak · · · a ~b · · · bn · · · 8& < R . & h+" ^+$ C ! %./ k5& CfX & B & =V+ lim rn 2j& r N+N C P T
KK !! '801

24 ^ =A = [; ] + i: 1 E < x ≤ x ∈ A P L & Ok< L =sup(A) = + x = L & G d Tx ≤ x ∈ A P L & S = ≤ S 9

n

+ i: < +

23 &4+ 2456 , -

KK !! 6 d < "& N: %! s ' =V+ ( Ǳ!) & d & O ;N :5 BC d $ %.! < M)N ǱD %! %.! +S L ! C & ; H , =V+

@+7 R N+N C @C. =>= @+A/ n< Z = H! R L +l < O& L @C. L P = .+ R L +l < ^+hH L @C. L P

& < O L & & < Td / VP H & < :S p 4 N =V+ | d %!& 3 T%De %.! < ! T< T' C & TǱD L <" & < 54 {+ & ǱD $ %.! & .C ^+.P =V+ ]S ." === T`0 T` C & $ & %! L 4 C < & 7+!< {+ ^ & =V+ ]S ." === < `> =%! " VP: Y+34 C W.

& 4 2.+N L ' P V+N & VP: ±¯a~¯b : & C C W. 8 T< W(& %./ & B 2.+N L ' P V+N & TmK! =V+ VP: ±¯a~bc : & C C W. 8 T< & M: k < TV+P t.P ^ =V+ +& N+N 3 & N T =VS S C @.P W. = P S C L +&

2 2

A = p ∈ Q p <

!

+P w C. L H! Q Q< T%! Q +l < O& L O& n< ww0x _+ <x w x Z Q T5 = = T"& (a−b)b > & S C b < a S +P w0

b¿ + a a+ b √

a − b¿ + b

√

b¿ + a a− b

a − b¿ b

=& P S C _+

+ T + S C L 8 a + i: √ √ √ √ √ √ √ √ T T < ( + )/( − ) T + √ √ √ √ & 7.B $ ' =%! + + −

L QD ' 8" & N+N C P 8 & B & Tw" B =9= [5 &x %" S C 5/< 8 & 8_ e %: N+N 3 & XN ^ T%S %+5/ ^ t d p C %! i:

+

w>

="& d @ a +&+& S & Y+34 {

" 4 r C C W. T%: d / " 8 }7& ' & V+hS

&

C @C. + '+D8 g 78 # = B< N+N 3 & U/< N @C. < N+N

N+N C = = ~ · · · < ~ = ~ · · · < C W. %! S /< < /< r ="&

Ǳ < %! 7+73 @!P D T4 ^ ! i: ="& 2+ +& {Q; {7X LD Q $

2 N

√

^ =A = {x ∈ Q | x < } + i: √ + A [5 L =%+ A AC sup(A) = 24 √ = + < %! 8$ L A L AC P S √ √ √ = ( + )/ > G d T > S , =sup(A) ≥ √ / "& A O& ' < %! L . √ = ^&& ="&

P4 5 < k && |( P 5

=%! m &&

S

−

~ < ~ C +P w =+& ! 8& d mK! < P

< ~ · · · C +P w0 = P S ~ ···

C P + %&E =%! 5+DX C n + i: w =%! %./ &/ n & V/ n &

T"& V/ P bi < P ai < ≤ n ∈ Z S +P w6

G d

=n = k− < %! P(Q 5+DX C k + i: w0 n T (Q 5+DX C n − P ^+& L + %&E n & d . . ( X C =%! %./ &/

n~a a · · · am b b · · · bk = a · · · am b · · · b k − a · · · am =n+

· · · · · ·

6

√

√ √ √ √

! "# $ %& '

≈ ≈ ≈ ≈ ≈

ln ≈

≈

~ ~ ~

~

=

√ π≈

≈

~

≈

~

ln ≈ log e ≈

ln

Γ Γ Γ

= ◦

=

%&E =abc ≤ < De N+N C c < b Ta + i: w> c a b = + + > a + b + c +

~ ~ ~ ~

~

b

< V/ %! ! + %&E w6 ! C W. T+ q B< V/ n! & V8 ^ d = B< 4 %! -+P V+.5 +

!" # $

~

◦ /π ≈ ~ π/ ≈ ~

n −n n! ~ ~ ~ ~ ~ ~ ~ ~

~ ~ × ~ ~ × n

=

x + xy + y

(x − y)

=

x − xy + y

(x + y)

=

x + x y + xy + y

(x − y)

=

x − x y + xy − y

x −y

=

(x − y)(x + y)

x − y

=

(x − y)(x + xy + y )

π/ ≈

x + y

=

(x − y)(x − xy + y )

π/ ≈

x − y

=

π/ ≈

x + y

=

(x − y)(x + y)(x + y ) √ √ (x + xy + y )(x − xy + y )

π

n(n − )

xn− y e

n(n − )(n − ) n− x y + ··· ! · · · + nxy n− + y n n(n − ) n− xn + nxn− y − x y

=

≈

e/ ≈ √ e ≈

+

(x − y)n

≈

π/ ≈ √ π =

V n 5+DX C < y < x (Q N+N C ǱL & xn + nxn− y +

%

%7C =VLH Z C $ :5 & & W(& ^
(x + y)

=

a

= P x+y+z = & N+N C z < y Tx + i: w9 =(x + y + z ) ≤ (x + y + z ) + %&E

V y < x (Q N+N C ǱL &

(x + y)n

c

n(n − )(n − ) n− x y − ··· ! · · · + nxy n− − y n

+

F

eπ

≈

πe

≈

~ ~ ~ ~ ~ Γ

≈

~

~ ~ ~ ~ ~

()*+ ,

&'()

TG +& & ="& Q & >=>= @+A/ n< Z =%! + ' < U.B ,.C .P & C
=

− i Tu

uvw

=

−i

2

S

G d Tw = − i

= =

( − i)( − i) = ( − ) + (− − )i

=

− i = − i.

G d Tv = − i < u = + i S

u v

C & & & ^+Q< & L L ,! 96) & '_ O<+S T( 2 c! & =]S " "d s7 ( "d s7 ( C & 696 ,! %! ,<

,. & 8 ^ L& T < =" "& + Ǳ+" ^ 7D& h: s7 ( C & 7.C @D!3 ^+Q< T \7.B ^ < @+ = 6I0 ,! %! S . 4f

!

u[vw] = ( − i)( − i)( − i) ( − i) ( − ) + (− − )i

=

2 N

C ^ B< \]H C < \]H ^+& . T Ǳ& i . _ +DO I)0 ,! < T & : 8& 2 f;4 & ^ =%" < ^ =P C < s7 ( C . P @;& /& 7< $ S_& + II) ,! √ = G" g − − = " ,NQ M . + ,X ^ & q P _+ SL! T
A/ < & S ?+P " 2D!3 5 & c
= ( + i) − i i + ( + i)

+ + + i ( + i)

− + + i = i.

=

u

= = =

v

G d Tz =

+

√

i S

2 N

√ √ z − z + = ( i − ) − ( + i) + =

+ D!3 L N L ' P ) ( − i)( + i),

!

) ( − i)( + i),

) + i , − i

)

−i ( + i). +i

24 &

)

)

!

(a + bi) + (c + di) := (a + c) + (b + d)i, (a + bi)(c + di) := (ac − bd) + (ad + bc)i.

V+ /

( + i) − ( − i) , ( + i) − ( + i) + i tan θ , − i tan θ

2j& < U.B ,.C C. ^ & =V+ [5 V+ [5 L

=

+ D!3 L N )

OP G!! @ C.

C = a + bi a, b ∈ R

%! s7 ( C w√+ i: w6 X b < a N+N C ww = − + i Tfkex + w + w = a + bw + ^++5 = − w w +w+

*

( − i) − ( + i) +

:= + i, := + i, −(a + bi) := (−a) + (−b)i, a −b = + i. a + bi a +b a +b

.

=+ %&E 0=F= @+A/ L w )x < wIx wJ

@+7

C

s7 ( C @C.

!

Girolamo Cardano½ Rafael Bombelli¾

I

()*+ ,

=|Im(z)| ≤ |z| < |Re(z)| ≤ |z| w

2

s7 ( C A ⊆ C @C. ! =+ ( |z − | + |z + | > < r4 T%! M75 A & z = a + bi + i: p ^ & = ^&& z

|a − + bi| + |a + + bi| > , (a − ) + b + (a + ) + b > , (a − ) + b − (a − ) + b + > (a + ) + b , (a − ) + b > a − ,

(a

− a +

a

+ b ) > a − a + + b >

[Q`0= 8" C. ^ =

,

+ i: 9:0 &4+ 2) '& , - ! =%! Q.5 @34 R = {(x, y) | x, y ∈ R} [5 C s7 ( C @C. < R ^+& L 2j& g %! '+D8 g ^ C a + bi → (a, b) ∈ R V+ s7 ( C mH =w" B [Q` = 8" &x C %! +Q ^+.P & = V R @34 N 5& %! d & T L +& = _+ s7 ( @34 (a, b) @;N & (, ) ǱD & & a + bi C

s7 ( C < U.B T S ^ & =V+G& 8 T j =w" B ` = 8" &x & P & < U.B t.P

.

a

+

b

+ =V . V+! >

s7 ( C U.B +D5 w T W. w[Q = 8" & U.B C &

> ,e 7e w

24 ^ Tz = a + bi + i: ! ! [5 a + b 2j& < |z| . & z a =V+ Re(z) . & < + z Im(z) . & < + z b =V+P z . & < [5 a − bi 2j& z =V+P =V+P

KK M0<: M0<: R S

,e A @C. w[Q 0= 8"

2 N

/< z , z , z ∈ C s7 ( C +P < β < α N+N C / %! s ' & /< < =α+ β + γ = < αz + βz + γz = " %: $ γ z < z < z N V+ B Tp ^ & = − z−→ z P& 5/< %! s ' & /< < /< z−→ " %: α N+N C 5 T"& L − z < − − → − − → z − z = t(z − z ) G +& & =z z = tz z

" i: %! : =(t − )z − tz + z = 24 & V8 ^ m8C & =γ = < β = −t Tα = t − ="& 2DE &/ & < z + z

+ z =

+ i:

2j Tz, w ∈ C + i:

PQ :

" !

|z| ≥ wM7; / &

x w

=z = S < S |z| = w0 =|zw| = |z||w| w> =|z + w| ≤ |z| + |w| we7e < x w9 =z − w = z − w < z + w = z + w w6 =

2 N

z w

=

z w

< zw = zw wF

=Im(z) = (z − z) < Re(z) = (z + z) wI =|z|

|z | = |z | = |z | =

= zz

w1

=z ∈ R S < S z = z wJ

fO < 7e ' ch z < z Tz +P 8" & = P wC : |z| = T5x < @ 3

=|z| = |z| < z¯ = z w ) 1

()*+ ,

P |<_ B < ^ D/ ^. && " B

=" B `0= S p f TI=F= @+A/ & B & =

!

|z − z | = (z − z )(z − z )

"& |z| = M7; / & z s7 ( C S + %&E w " %: t N+N C G d Tz = − _+ < =z = +ti −ti

= z z + z z − z z − z z = =

L C/x + %&E TP(Q z, w ∈ C + i: w0 =|z + w| + |z − w| = (|z| + |w| ) wfO =+ %&E I=F= @+A/ L w

d @74: mH T B< N

a + b +

+& s&< r4 P z @.P @C. T P ) |z −

) |z −

+ i| = |z +

| + |z + i| = ,

) |z −

| ≤ |z + |, z+ ) z − i = ,

)

− i|,

) |z − i| = Re(z) +

a + bi

+P T|α| = |β| = |γ| = + i: |βγ + γα + αβ| = |α + β + γ|,

)

(β + γ)(γ + α)(α + β) ∈ R. αβγ

+

b y

c = . z x∈R

− α ≤ a ≤ + α, b=± α + a −

−a

.

2+ N

=

= P (z) = an z n + · · · + a z + a

= an z n + · · · + a z + a = an z n + · · · + a z + a = P (z)

TBC & ( F 'U '8( D 2# N

V+ i: 2j = = b − ac < < ax + bx + c = ^&& =(x − ba ) = b −aac Ta(x − b ) + c − b = a a

i: & L 2O5 w 1

) (x + i)n − (x − i)n = ,

)

− a b + b − a + a b = α

{ & 7.B $ ' P (x) S +P $ ^ @ _+ z G d Td L ' z < "& N+N =%! 7.B n P (x) = an x + · · · + a x + a V+ i: =

G d T"& P (x) @ z S = N+N C P ai

V+S |<_ ;& ^ ^+:X L =P (z) =

< ch z < y < x _+ < c < b < a S +P w I 7e < ^ 4 G d T"& R = C 7e & a x

^+& $ < = && <&<

z

5 T%! ,D/ + mH T N+N C b < a $

%De ,8 L & =b = ± α + a − − a ^&& =(a − ) ≤ α T +a ≤ α + a 5 T"& %! ^+$ Q@ B

) |z| − Re(z) = ,

)

<

(a + b + ) = a + α b = −a − ± α + a

) |z| + Re(z) ≤ ,

|z − | = | − z¯|

z

α = |z − | = |a − b + abi − | = (a − b − ) + (ab) .

< |α| < S +P w6

G d T|β| ≤ -" /& < 4 $ =|β+α| ≤ | +αβ|

+ i| < ,

<

+ ( P z = a+ bi ∈ C @.P @C. =|z − | = α

G d T r4 " ^ z = a + bi S =

+P T|α| = |β| α, β, γ ∈ C + i: w9 =|α + γ| + |α − γ| = |β + γ| + |β − γ|

) |z −

z

= .

2 N

x < wFx Tw>x w>

) Re(z) ≥ Im(z),

|z | + |z | − (z + z )(z + z ) − |z + z | = − | − z | = − |z |

b ± x= a

cos x + i sin x = sin x + i cos x.

J

b − ac b ± = a a

ac − b a

i.

()*+ ,

)

(reiθ )n = rn enθi ,

)

reiθ

= re

−θi

)

V T

+i=

( + i)

=

iθ −

)

, √

eπ/ $

+ %&E T%! P(Q 5+DX C α + i: w J

− reiθ = rei(θ+π) , (re )

2

=

r

e

−θi

+ i tan α − i tan α

,

=

=

= ( )

4 < ' [Q( s7 ( C b < a + i: w0 < a < TO ch & 7e < +P ="& = P & ab < b TO ch & _+

√

eπi/

+ i: 9:0 &4+ 2;< , - # ! @74: =%! 4 [Q( s7 ( C ' z = a + bi ∈ C Oz s(.+ < Re 3 ^+& %De @<L < r ǱD z @;N V+ θ

( + i)

! V8 @." & TP_ H C T8 Y+ = " 0=F= @+A/ L

−i=

√

eπi/ <

+

√

i = eπi/ $

= ( )

=

a ⎧+ b . ⎪ arctan (b/a) ⎪ ⎨ π/ θ := arg(z) = ⎪ π/ ⎪ ⎩ arctan (b/a) + π

r := |z| =

2 N

√ + i eπi/ = √ −i e−πi/ √ √ () = e(πi/+πi/) (=) ( ) e(πi/

)

! V8 @." & TP_ H C T8 Y+ = " 0=F= @+A/ L

|z|

z = a+bi Re(z) Im(z) − i −i +i −i − +i − −i

sin θ + sin(θ) + · · · + sin(nθ)

θi

S S S S

z = reiθ

V [5 & B &

2 N

= Im(eθi + e

z = r exp(iθ)

%!& sin θ+sin θ+· · ·+sin nθ 2DC N V T 0=F= +A/ L w x %./ L ! & =V
θi

a> a=b a<

e(π+πi/ ) (=) e−πi/ cos − π + sin − π = − i.

= Im(eθi ) + Im(e

+ i tan(nα) . − i tan(nα)

=

%&E 24 ^ T(cos α + i sin α)n = + i: w0) =(cos α − i sin α)n = +

!

() √ = e π/i √ √ e(π+πi/) (=) eπi/ √ √ cos π + sin π

n

) + · · · + Im(enθi )

+ · · · + enθi )

()

= Im(eθi + (eθi ) + · · · + (eθi )n ) − (eθi )n θi ( ) − enθi θi = Im e = Im e − eθi − eθi enθi/ (enθi/ − e−nθi/ ) θi e = Im eθi/ (eθi/ − eθi/ ) nθ () (n+)θi/ i sin = Im e i sin θ sin nθ n+ () . = sin θ sin θ

−

−

−

− − −

√ √ √ √

s7 ( C D;/ W.

! V8 @." & TP_ H C T8 Y+ = " 0=F= @+A/ L

)

reiθ =

)

reiθ = r cos θ + r sin θi,

)

cos θ =

) 0)

(r e

iθ

eiθ + e−iθ

)(r e

iθ

arg(z)

π π/ π/ π/ −π/ π/ π/

z = reθ ei eπi eπi/ πi/ √e πi/ √ e −πi/ √e πi/ √e eπi/

!

%! L n" & Z

r = r , θ = θ + kπ, k ∈ Z,

) reiθ = r eiθ ⇔ ⇔ r=

!

< θ = kπ ,

),

)

k∈Z

sin θ =

) = (r r )ei(θ +θ ) ,

eiθ − e−iθ , i

()*+ ,

Tnθ = θ = kπ < r = rn V+S √ =θ = (θ + kπ)/n < r = r ^&& √ G d Tv = e πi/n < u = reθi/n V+ i: S %"

=%! (Q k

∈Z

!

n

T−

n

√ n re(θ+

kπ)i/n

=

√ n reθi/n e

πi/n

k

=

=

√

T(

− i)

√

exp

+ kπ i

+&+&

= e

= cos

π

+ i sin

π

√

sin z =

i

^+t.P =z =

z

z + + z

z + +z

= = = = =

z = + z

z + + /z

=

z + +z

<

<

cos z =

zi e + e−zi

+ %&E 24 ^ Tsin z + cos z = w[Q < tan z = sin(a) + i sinh(b) w cos(a) + cosh(b) =sin z = sin a cosh b + i cos a sinh b w| √ n

+ D!3 L DC L ' P . w1 < cos x + cos(x) + · · · + cos(nx) w[Q =sin x + sin(x) + · · · + sin((n − )x) w

z = + /z

z z z z + + + z+ +z +z z + z + z +z +z z + z + z + z ( + z )( + z)

(

5+DX C ^ 8$ w6 √ m = + i = ( − i)n

n

5 Tz z = L %! 2DC

z

−i +i

+P Txn + iyn = + i 4 wI √ < xn yn+ − xn+ yn = n w[Q =xn+ xn + yn+ yn = n w

5 =zz = ^&& Tz = $ =

O& < ,< vX T^&& =z z + +z

<

zi e − e−zi

2 N

z + +z

i: & 24 ^

√

N L ' P w0

ez := ea ebi = ea cos b + (ea sin b) i

i =− − V Tk = &

V+ [5 Tz = a + bi ∈ C S wF

π π i = eπi/ = cos + i sin =− + T%! ' VH P L 8 z = + i: + %&E 24 ^ √

m

ei = cos + i sin =

π/

√

+√i − i

G d Tθ ∈ R S +P w> Tsin θ = sinθ cos θw[Q π < T sin θ = sin θ sin − θ sin π + θ w π π sin θ = sin θ cos θ sin − θ sin + θ w| + %&ETn ≥ 4 w9 π sin n sin nπ · · · sin (n−n)π = n/n−

V Tk = & √

T

=+ D!3

V Tk = & =k = , , d

√

√

n

2

ei =

√

T− + i T− C L ' P w +cos α+i sin α =+& D;/ 8" & +cos α−i sin α

√ i √+ +i

^ n = < z = + i: ! ! & && ' C ! @ ! T24

√

√

= uv k .

fk.C uvn ^&& Tun = e πi/n = e π = :X L < 4 ^+& N k %! : mH =%! uv = u .P 2 = + n −

√

− i

< z = reθi ∈ C − {} V+S ) =/& ! L DC z C n @ n 2j =n ∈ N √ n

z=

√ n r exp

θ + kπ i n

=k = , , , · · · , n − d

< w = r e Tz = reiθ + i: 2DE V T 0=F= @+A/ L wIx %./ && < z = wn ^&& + 0=F= @+A/ L w x %./ L =reiθ = rn einθ

+ z + z + z )( − z) ( + z )( − z )

√ =w = n z

(z −

) = . (z + )(z − )

0

iθ

()*+ ,

%! s P wGS j Tx . %"G L ! & S & < & %! ; =" "G S & P < C ; . mH = ]S N L " j . & wG x @;N ^ ! ;N ' & %+& & ;N ^ =w" B `>= 8" &x V+P ∞ g N & C∗ = C ∪ {∞} =%! g N ∈ S Z ^ . = T%! L DC ∞

" !

T

) z∞ = ∞

√

√ − √ cos π + i sin π √

T − + i T i T

D!3

N L ' P w <

√

+ i =+

=+&+& >0 VH P . w0

' V W <; OP G '86B7

)z+∞=∞

√+i −i

√

P mK! < " D;/ 8& − i C w> =+&+& d $

) w = ∞ w ) =

=+ D!3 (

∞

' n P = =w + w

%+& & T^&& =w ∈ C∗ = C − {} < z ∈ C d

= s7 ( C .P ∞ s7 ( C %Q & %! ^ " :S p & 8 ∞ Y+34 fk 2D!3 24 ^ =V+! < VD ∞ P

− i)/

[7 ( N . w9

S +P w6 + · · · + wn = G d T"&

w , · · · , wn

L 2DC +P Tm

∈ N

< a ∈ R + i: wF %! x x −a && −a m

m

π x − ax cos +a m π × x − ax cos +a m (m − )π × x − ax cos +a . m √

G d Tw = − + i S +P wI

(x + y + z)(x + yw + zw )(x + yw + zw) =

@34 \ S w

= x + y + z − xyz

GS %"G w[Q >= 8" ∞ <_: & s7 (

=+ a ∈ R x − ax + (a +

!

> ?$@ 9:0 AB8

) = @Q5

w1

C ) % ' !

& _+ R
8" & ! @B DB @Q5 + i: w E : X + AX + BX + C =

= P N+N C d { T%! " X = x − A/ i: & +P 24 ^ D E : x + bx + c = ! : & E @Q5 +P < x = s + t + i: , =. v] & T =s + t = −c/ < st = −b/ & s { & < @B Q5 & TO& @Q5 < ^+& t E &B T d %!& @Q5 & TmK! =+!& =" B & =+&+&

S : x + y + (z − ) =

&x "& S ,." {;/ N (, , ) + i: ^+t.P L x+iy = (x, y, ) ;N P ǱL& =w" B [Q`>= 8" 3 < :S p N < ;N ^ ^+& 4 < %! s TC ^ & =V+P f (x + iy) . & S @ & d /f ^ =d %!& S − {N } & C L '&' "G {+ S < %"G

XX

I0 $8

C < N = c + d < M = a + b S +P w0 5+DX C < L C. 24 & "& 5+DX C. 24 & _+ M N G d T " "

"0 M@"

f : C −→ S − {N } f (x + yi) =

00

+x +y

x, y, x + y

- . / 0*1 2

2DC .Px =%" 5+DX C < L w=+G& p |(a + bi)(c + di)|

^+.P & < d & & L .+& 5 w1 =%! . Q. H 2f8 +Q Q< G %! _: c! K+ & =. X 78 +! : : < d B< & & + d \ S , +SR< < t( @p f 5B % @8D" & http:\www.maplesoft.com cd =+

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n; ^ L O & " D " L ! & : +Q & P . P 7O Q T"& \Ld K+ ! p & 8 D & " +S L < V75 Tc w =" v5 +( P_+$ d '. & c w0 {7; V+ G G$ < +& +C & S = U L 4 d %!& %4: L c w> +& 2. < LK& {Q; M.C & TK+ +S8& = cf B 5! Q. P w9 =" U7; d ! G G$ < .

< 7e < B B B < A A A + i: w> + s7 ( C d chx P(Q fAQ Ai Bi s H s!< Ci + i: =w+G& p _+ C C C 7e + %&E =i = , , T%! =%! fO < =%! N+N C a < 5+DX C n + i: w9 2DC N s7 ( C '.8& T24 ^ n

sin a +

n

n sin(na) sin(a) + · · · + n

=
+& 7C ^+jj( + -/ED " <: 2e D "Ld '. P_& +S8& _+ < 3 Q< QNQ M d B< 4 G.P B< = P < 2p d L ! G G$ < _+ T n; P_: <3 @ ! ^N 8 " < ,. TP_: L S +S 8& & %! ^ 8 ^ =< Pg < & H < {7; M.C L V75 U &+"L < m @+" & ! M & p& 8" & %! & c! ^+.P & =. & 7.C P8P < . M+N3 Zj ^ D =. h V ^ L 28 & 3 @ " +4 c! ^+.P & "& " : B s+3 & L K+ _: & "d & w Q< + d '. & _+ ! ,e $ < " "d d 7B & +P %4: < + +& %/< v4 L =<& ;& wwp 3&xx & Z P w0 wwK+ L !xx W(& & mK! < +P / B =+ 5B T%! "
+, - ./(0

1

L =%! / +& +K _: ' wMaplex K+ & < M+N3 T\Ld S 8" & _: ^ %! .+& _ +K =" ! T8 T{7 T8+ . T+p x & P_: ! L d n" & +SR< ^ L & =L! _. w= = = < L = d Y+34 C & 2D!3 w = V/ 5 P & C 2D!3 w0 = d '. & ^. 2D!3 w> &L & .+& _: P & < " ! U& w9 8& Zj(& C P B< K+ =d 25: < sD D3 S P w6 =. ! 5& +K {! ' C & d s+3 L wF =. ! , ^+C < / +& & &L ' K+ wI =%! ! 0>

- . / 0*1 2

=+ +u

d mK! < %! & & +Q< Qe + 5! w> =+. K+ '. & 96 P P Tc < @ P L 5& " +4 w9 d < " :S p w GLdx C & N+/ (&! _+ < ! G G$ \Ld & K+ & s7 ! =LK& d & .P 7 X& c " +4 w6 = {7X S" L Vp X & K+ L ! & < !

& L ,<B 2/ FG/ ( 6& - ! " %! " u d ,5 < 2+ ,.5 P. L

Q.5 ^ K+ s+3 a+b a−b ab a/b ab sin cos tan cot sec csc sinh cosh tanh coth sech csch arcsinh arccosh arcsin arccos arctan arccot (x) ln(x) [x] |x| n (x) max(x, y) min(x, y) log (x) logn (x) π √ i = −1 Re(x) Im(x) x¯ /x sgn(x) ex

a+b a−b a∗b a/b aˆb sin cos tan cot sec csc sinh cosh tanh coth sech csch arcsinh arccosh arcsin arccos arctan arccot sqrt{x} ln(x) f loor(x) abs(x) root[n]{x} max{x, y} min{x, y} log (x) log[n](x) Pi I Re(x) Im(x) conjugate(x) /x sgn(x) exp(x)

@ %! LO K+ L ! & I 5*J " "d ^. TX& "Ld { L 8 & 5B & 3 G G$ < '. L ! $ DQ; TK+ s+3 & =+G&: d {Q; K+ s+3 ! P C= (K ) /& " < &/ & s! %! P. < v< L QD

! q & < P. L & ! P ="& d @+ < " B ! T" ! q . L S =" ! ' d . S Q< TS Pg 5& s

W. < " B p: ! d T" ! =" P( d

L ' %6 (& 2 I %& B I M "

& < / ! Zj( X & ." G ! =+ B Setup : T<& =. P {j K+ _: & T5x K+ 8" d & ^8d ' T{j L mH & <" & =S Pg T%! & WN w| % ' ^ L mH =+ '+7 & < ^8d d & %! :

d O& < $ %.! @"S S Pg +! @34 %! : 2 ! . < & T/ _8.$ ! ' P d =+ & <" < '7+7 ] @34 & =S fC + < " B ! TL Enter & ! +7 T : Shift %+" +7 Q t $ < & B s ' T" B ! 8 <& T+ _& =" L& 7D/ 2 ! @

%! : T ! B Qe L ! & < : {+ & Open < File P+7

P , =<+& ! Examples\Volume_1 wworksheetx 34 & < " p 7j: L ' =+ '+7 p ,e " + T Qe 3 ++a t $ w / ." G ! x G @+ Maple7

& < B C ! + i: ( )H@& - " " %" V+P 24 ^ TV+"& +! R @+ K+ C −−−−→ R

09

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m

C %Q 9:0 &4+ G + " x + s7 ( C . < & =%! s7 ( K+ s+3 √ S =x+y*I " TK+ s+3 yi = x + y − %! : TV+P %Q D!3 x N+N C V+P(& =V+ B assume(x,real) !

,X z = reθ C z Sd z D;/ W. z <<

n

V+ i:

NA8 &4+ G + "

24 ^ T"& 5+DX C

Q.5 ^ m @ .+H & n n C @_ -%! ,< C n d

K+ s+3

} +7C N ^ 8$

^ S_&

n < m } A n :

( ,< C ^+ n m n L e(k) . n m

K+ s+3 Re(z) Im(z) conjugate(z) abs(z) polar(r, theta) argument(z) convert(z, polar) 1/z

n mod m factor(n) isprime(n) gcd(m, n) lcm(m, n) n! binomial(m, n) ithprime(n) sum( e(k) , k = n..m)

C m < n V+ i: /@ &4+ G + " V TD/ 2 ! &
< Q5 P & & 73 " TV+ < K+ s+3 V! ' & Q5 ]H 24 eq_name:=equation ! & ^ ="& Q5 @;& equation < Q5 eq_name @Q5 eq_1:=x^2+y^2=1 eq_1 & x + y = = :5

!

ifactor(n) simplify(n) numer(n) denom(n)

23 &4+ G + # "

i: N+N C +P(& t $ < " #$%& '( #) =+ B with(RealDomain) ! & & " &
2O5 " 2O5 G ! V+P(& t $ ! L %! : TV+ eq_m < === Teq_1 " ! solve({eq_1,...,eq_m},{x_1,...,x_n}) = P Q@ 2O x_n < === Tx_1

cd

<

Q.5 ^

"

http://webpages.iust.ac.ir/m_nadjafikhah/r1.html

=%! "
06

K+ s+3

m k @ root[k](m) n @ sqrt(n) n @" ! simplify(n) n @" L& expand(n) V/ m & n C W. evalf(n, m) n @" S 24 rashnalize(n)

- . / 0*1 2

0F

2

√

L T%+ U& yx = y S sN: < S x → xz g =" g y = − < y = jC < & x =

< + ^ 74 C L mH U& < + L .C %./ TU/< =%! ,G [5 Rn → Rm : & U& @ 5Q; V7C ,G & & T3& %Q3! & %! ^"< T DQ q. 7 %Q < =VLH m = n = ! %Q @5Q;

"d j: ^ L vP =V+ 5Q; < 7B T%! R → R 8" & 5& 5 TU& L 4 & U& I j: = +S / ! 5& 25Q; =& P U& ^+.P @ !

3 '8C!

=f : X → Y + i: f (x) d L & P x ∈ X @.P @C. L %! 2DC " [5 f

Df := {x ∈ X | f (x) ∈ Y }

x ∈ Df %! P f (x) @.P @C. L 2DC f

Rf := {f (x) | x ∈ Df }

&!

4 < Df = X V+hS 4 f U& f U& 4 =Rf = Y V+hS < Tf (x) = f (y) < L x, y ∈ X P ǱL & V+hS =S + x = y

=).

234 5!4

@

4C PG ! %! G ! ' " p L U& & Zj(& +7.C mH < :S 3 C. %! ^+$ U& .! [5 = fC + Td

2

,e %P =R = X = Y + i: !! < & "H T& '+D8 Z P L " Z ^ L 8 U& 5 =7N & "D 8 < "& G Z 8 L N T"& f (x) x log x

Df R (; +∞) x R √ x [; +∞) x + x R tan x R−{kπ+ π } x R /x R − {} x +x

Rf R R (; +∞) [; +∞ R R [; +∞) (; +∞)

=P(Q @C. < Y < X + i: Y < X ǱAC ^+& %! g TY X L p %D Y L AC ' e X L AC P & S & f : X → Y V+ TV+P f & g ^ S =P s! x ∈ X AC S = Y X f V+ < . & < + x f y T" g y ∈ Y & f & x → f (x) ]S. L =V+P f (x) =" ! f @;&

' >! =

'+D8 "H !! √ √ √ √

√ √

√ √

√

M! ' >! Ǳ!>! ' !K

√ √ √

24 ^ =R = X = Y + i: =f () = Tfke =w .P U&x %! U& ' x → x g k e =%! U& ' x → x g f () = Tf ( ) = Tf f (−x) = f (x) ≥ U& ^ =f (−) = <

2 2

2 N

@;& & f : R → R + i: '+D8 f d + ( Tf & < ^++5 ^. =%! -%!

f (x) =

0F

345 6 # 5

) f ) f

x+ x

345

x

=x +

=x+

x

5 T" [5 x+x x ∈ Df /< < /< = =Df = R − {− } L %! 2DC f @ TmH = + x = f (x) = y @Q5 =y ∈ R V+ i: f & ^++5 &

(|x| ≥ )

,

+x ,

(x > )

2%

^&& =x

"& (Q N+N C c < b Ta S +P

G d Tf (x) = ax + bx + c <

x

f (x + ) − f (x + ) + f (x + ) − f (x) =

x V+S +x y ± y + y y ± (y + ) −

− yx − y = = =

' I ⊆ Df < f : R → R V+ i: ! ǱL & %! I f V+hS 4 =%! L& 4 =f (x) ≤ f (y) V+"& " x < y x, y ∈ I P I f V+hS x, y ∈ I P ǱL & %! & 24 & =f (x) < f (y) V+"& " x < y

4 U& =S [5 < 4 d q"& Q<_ < 54 V+hS %! ^"< ="& Q<_ k+ < 54 k+ V+hS =P + & '+D8 T8 k+

&7

( S" Y! ! ( S"

" U& 8 @ T P

< Tx

xy + y = mH Ty =

V+ (

& =f (y)

2 N

− x U& =V+G& p f (x) = < y 24 ^ T ≤ x < y ≤ V+ i: = ^&& = − x > − y ]Q f (x) =

−x >

< x P ǱL & f : R → R S x =+ ( f @;& T r4 f

"

p

x G d T|y + | ≥ S 5 T(y + ) − ≥ S mH %! S ^ & ,5 " ^ T =f (x) = y %P T5 =y + ≤ − y + ≥ L %! 2DC f & T^&& =y ∈ (−∞; −] ∪ [; +∞) T%+ '+D8 f U& T
&7 Y! !

!C. !C. Y! !

T

x x+

=

2 N =x

x+

V+ i: p ^ & = y −y

^&& =x

y −y

=

Tx

" B =f (x) = x V
-$ =%!

− y = f (y)

L 2j&

V+ i: S =%! Q<_ k+ I = [; ] @L& & f T5 =f (x) < f (y) ]Q < y < x 24 ^ T− ≤ x < y ≤ & B & =%! 54 k+ I = [− ; ] @L& & f U& 5 =%! . TDf = [− ; ] 8

)y=

4 + ( f @ T"& " :5

y = f (x)

x − x ,

) y =

) y = (x − )

2 N

=V+S p f (x) = x + U& x− !& f (x) < f (y) " = < x < y + i: = T P %De x − < y − $ = x + < y + V+ x− y− V+ (x − )(y − ) vX < V+

) y = sin

(x + )(y − ) < (y + )(x − )

) y = ln(

√

x+ , −x

√ sin( x),

) y = log(x

− ),

) y =

x,

|x| − x

.

+ ( y = f (x) & < T P

) y = + x − x

=y < x < Txy − x + y − < xy − y + x − mH %&E & 2j& =%! Q<_ k+ I = (; +∞) & f mH & T =%! Q<_ k+ _+ I = (−∞; ) & f " ' P & f V+S + TDf = R − {} 8 & B =%! Q<_ \ P L& L

,

− cos x),

) y = (− ) y =

)x , x . +x

P '+D8 L y = f (x) U& L '+

) y = x − x ,

) y = x ,

) y = x + −x , ) y = x − −x . +&+& 4 y = f (x) U&

0I

345

345 #4 "

f (x)g(x) = (x + )( − x) = −x − x − x + ,

( L U& 8 P

(x + ) , f (x) = = −x − − g(x) ( − x) x− f (g(x)) = f ( − x) = ( − x) +

= x − x + ,

g(f (x)) = g(x + ) = − (x + ) = −x + , f (f (x)) = f (x + ) = (x + ) +

g(g(x)) = g( − x) = − ( − x) = x.

2 N

=

x

=

/x

:= = =

f (x)

:= =

2 N

/( − x) + i: f (f (x)) = f −x −x = − /( − x) −x− x− = − x x f (f (x)) = f − ( − /x)

− =

/x

=x

= f (x) = f (x) = · · · = fn+ (x) =

f (x)

= f (x) = f (x) = · · · = fn+ (x) =

f (x)

= f (x) = f (x) = · · · = fn (x) = x

) y = ax + b . cx + d

f + g : x → f (x) + g(x),

f g : x → f (x)g(x),

f (x) f : x → , g g(x)

01P7 f + g Tf a 4?P7 af f M0<: RZ f /g Tg f 4?P7

f g Tg f f ◦ g O& < g

−x −

af : x → af (x),

f ◦ g : x → f (g(x)).

x

Df +g = Df g = Df ∩ Dg , = Df ∩ Dg − x ∈ Dg g(x) = , = x ∈ Dg g(x) ∈ Df = g − (Rg ∩ Rf ).

Daf = Df Df /g Df ◦g

Tf (g(x)) U& TL L ' P +&+& g(g(x)) < f (f (x)) Tg(f (x)) /x,

+ bx + c,

^ Ta ∈ R < 5& g < f + i: V+ [5 f ◦ g < f /g Tf g Tf + g Taf U& 24

x

f (x)

) f (x) =

) y = ax

T ! U& _+ < d P d n" Q.C L ! & ! +& \< ^ =
V+ [5 < "& P(Q 5+DX C n S T <

G d T fn (x) := f (f (· · ·!f (x)))" & n

) f (x) = x ,

) y = sin(x + π).

234 !3

Dg◦g = R − {}.

= ,

24 ^ =f (x) = f (x)

) y = log x,

∗

) y = (ax − b)(cx − d)

Df ◦f = R.

x

) y = x ,

, Dg◦f = R − {}.

(f ◦ f )(x) = f (x ) = x , (g ◦ g)(x) = g

) y = x − [x],

Df ◦g = R − {}.

, x x

) y = x − x + , ) y = − x ,

∗

p g(x) = x < f (x) = /x @! U& < Dg = R TDf = R − {} 24 ^ =V+S

(g ◦ f )(x) = g(f (x)) =

+

L U& 8 P d < c Tb Ta [7 ( N ǱL & + (

= x + x + ,

(f ◦ g)(x) = f (g(x)) =

<

g(x) = x + . g(x) = x + .

2

@! U& 24 ^ =V+S p g(x) = − x

f (x) = x +

f (x) + g(x) = (x + ) + ( − x) = x − x + ,

01

345 #4 "

345

$ < < L ! & (
%!& fn (x) U& Tf (x) = x/ + x 8+4 w0

S [5 0=0=0,e L w>x %./ fn x =
√

−x x< S x +x ≤ x S √ @;& & [ ; ∞) @L& & < − x @;& & (−∞; ) @L& &

( f (x) Tf (x+

f (x) =

4 f : X < Dg = Rf @ & g

$ U& [5 & X" =%! S [5 x +x T5 T"& SL! P;& %! ^ B< ;& N d G d T" [5 N < & x = x ǱL & f S ="& && U& & < & f (x) =

x− x

x< ≤x

) = x − x+ 4

=+

→ Y

U&

w>

P/Q

\].

5& V+hS S %: $ Rg = Df &

: Y → X

) ∀x ∈ Df : g(f (x)) = x,

) ∀y ∈ Dg

#

y = f − (x)

S S

=
: f (g(y)) = y.

. & < + f

^.

g %Q ^ =P

U& ' ]K!85 ^++5 +& & = B< L n" & ! +A/ Ti< f : X → U& 8 & : < LO "

="& '+D8 %! ^ T"& ]K!85 Y && W!85 U& & & U& P @ ! c85 =%! && W!85 U& @ & U& P & =%! TQ<_ U& P c85 =%! 54 5& T54 U& P =%! '+D8 5& T'+D8 U& P c85 =%! Q<_ 5& =%! && U& & TU& P c85 c85 ( f (x) =

x + U& c85 x +

"

=+

=Df = R − Ok< V+ B p ^ & = < xf (x) + f (x) = x + G d Tx = − S c# f (x)) ^ =x = ((−f (x)− ^&& =(f (x) − )x = − f (x) )

Rf = Df − = (−∞; ) ∪ [ ; +∞)

L U& L ' P & < B< 24 d L ' P c85 mK! < + ( +&+& ⎧ x< S ⎨ x ) f (x) = x ≤ x ≤ S ⎩ x < x S ⎧ x ≤ S ⎨ x+ ) f (x) = ⎩ x + < x < S (x + ) ≤ x S ! =". <3 $ ^+.P & U& [5 \< =" P u k5& < P +& p 2N "< k e f (x) = + x + x + · · · 2j& 5& %! ^8. Tf V+ [5

−

Rf = Df− = R −

24 L U& L ' P c85 )y=

x− , x +

) y = x/ + 0J

) y = ,

) y =

+&+& B< x , x+

x − x .

345

345 7 8

^+t.P < Df x f (x)

= [; ]

S p f Tp ^ & < Rf = [; ) ∪ (; ]

/ /

234 67

!0"

f : X → Y i< [5 Γf := {(x, f (x)) ∈ X × Y | x ∈ Df } 2j& R & X × Y mH TX = Y = R {7l $ =V+ = R @34 Γf N < %:S 8

/ /

=S 4 ` =0 8" ^&& + V! L U& L ' P . )y = x− ,

) y =

) y = x − x

√ cos x,

) y =

) y =

[x]

,

,

) y = arctan(x

,

x −x

) y =

V+! \< ^ ! 2G <- ? ) $ \< ^ =&;N \< & ! %! "< TU& 34 (x, f (x)) N mK! < ( Df L ;N ^ =V+ 4< V& N ^ T G d =V+ ( 7+Q $ %+ M+/ B< ?+P & Q< T%! ! +& \< -"& j i< @;N < ^+& U& . +& < -+ %! ! d T& j 24 L ' P & !H & < %! nX &/ G 2O! V+P Tk5& ="& U+!< 2CfX & L+ T d 5 < %! d G !+H 5 & TU& ' . & j = G & f < f P4 5 & _+ f !

) y = [sin x],

) y = x − [x],

),

x+ . x−

+ V! ;& $ U& L ' P ⎧ ⎨

) y = ⎩

x

x≤− − <x< x≥

/x

x ⎧ √ ⎨ x+ arcsin x )y= ⎩ √ x−

S S S

f (x) = x

) y =

x<− − ≤x≤ x>

f (x),

) y =

) y = −f (x), " y

= g(x)

f (x)

−

[5

/ −/

/

U& . 4 + V! L U& . TV+"&

= f (x)

) y = f (g(x)),

) y = f (x)g(x),

) y = f (x) + g(x),

)y=

f (x) . g(x)

< y = f (x)f (x − a) 4 f (x) =

2

4 N < . V+! R N ^ %! : =w" B [Q` =0 8" &x + j V&

,

) y = [f (x)].
U& .

x − − − f (x) − / − / −

) y = |f (x)|,

(x),

/( + x)

=+ V! V 9= =0 ,e L w0x %./ = & Q<B =Rf = (−∞; −] ∪ [; +∞) < Df = R − { } V+ + L n"

S S S U& . & 75 & B & L U& L ' P . + V! y = f (x) ) y = f

− |x|

|x| ≤ |x| >

w[Q =0 8"f (x) = x

2

S S

/( +x)

0 ,e U& . w U& .

V+ V! L U& . ⎧ ⎨ x

+ V+! Q y U& . =a = w| < a = w Ta = w[Q

f (x) =

>)

⎩

x+

≤x< x= <x≤

S S S

2 N

345 7 8 7 975

345

x + = ln x−

= − ln x − x +

= −f (x)

234 67 6 864

IK 4 I ⊆ R @C. L ="& M75 I & _+ −x T x ∈ I P ǱL & V+hS R > 4 %! N \ y = f (x) U&

+ ^++5 L U& & : < |
) y = x sin

) y = sin x + cos x, ) y = x

x,

P ǱL & V+hS 5& ^+$ . =f (−x) = f (x) V+"& " x ∈ D B [Q`0=0 8" &x %! N P y 3 & %D =w" 4 %! N \ y = f (x) U& P ǱL & V+hS 5& ^+$ . =f (−x) = −f (x) V+"& " x ∈ D =w" B `0=0 8" &x %! N ǱD & %D

− |x|, − e + x

) y = arcsin x.

) y = x e x

) y = sin x,

) y = sin(sin(cos x)),

,

V

) y = x + sin x.

2_

- B< "& : VP < |
4 C

4 C

: U& ' . w |
2

=%! |
∀x : f (x + T ) = f (x)

2 2 2 2+ 2# 2%

"& T < & < < 5& y = f (x) S = r4 8 & G d TV+"& V! [; T ] @L& f . < L G 25;/ TT A @L & < P x 3 ! =w>=0 8"x V
y =

f (x) + f (−x)

<

y =

f (x) − f (−x)

f = 24 ^ =%! : 5& y < |
< U& ' . >=0 8"

f (−x)

! < T < & y = f (x) + i: T4 [Q( C a = < %! S < & y = g(x) 24 ^

2

= (−x) sin(−x) = (−x)(− sin x) = x sin x = f (x)

L T%! : f (x) = ln x − x+ (−x) − = ln −x − f (−x) = ln −x + (−x) +

=%! T < & 5& af (x) w >

U&

2 N

345

345 7 8 7 975

2+ N

@< V

x∈Q x∈ /Q

=%! "& De C ^ S_&

S S

S 24 ^ ="& S C T + i: _+ < + < %! S _+ x + T G d T"& S

< & S C S N

<

S T T N

S w>

f (x) < f (x)g(x) Tf (x) − g(x) Tf (x) + g(x) &< g(x) t $x ="& d L ' P < L &A N < S w=&< ] U& T"D S C

2

T

T%! π < & y = sin x U& " x U& + U& < %! π = × π < sin ="& π < − sin x

< %! S _+ x + T G d T"& S x ∈ R S x ∈ R P ǱL & TmH =f (x) = f (x + T ) = ^&& & (Q < T ∈ Q V B Tf (x) = f (x + T )

2 N

2

^ =%! π < y = cos x U& = P π < & − sin x < cos x T24

π +P ="& y = sin x U& & < ' . < U& L ' P < @< T&< L U&

< & 5& f (ax) w0

U& L ' P G d T"& 5+DX C

f (x) = f (x + T ) =

N

T |a|

2 N

) y = cos x,

) y = sin(πx),

) y = sin

) y = tan x + sin x,

%! π < y = tan x = sin x U& cos x O& @+A/ L w>x %./ " :5 N C Q " B =%! π && y = cos x < y = sin x U& =π = × N

) y = x −

<

) y = cos

x, x−π

,

) y = arctan(x), ) y = x

x

,

) y = | cos x|, ((x + ))

+ sin x.

i< C & Y+34 C ^ 8_ f (x) + i: w + %&E Tf (x) U& @;& ^++5 ^. ="& x ∈ R mK! < %! < f (x) − x Q< %+ < f (x) =+ ( d <

2 N

Ǳ_B U& Ok< L T%! T

=

(x + ) − [x + ] = x +

=

x − [x] = ((x))

=

− ([x] + )

T((x + T )) = ((x)) x P L & < < T S T E < < ((T )) = α S L T"& 5+DX C T & G d @D!3 & B & < T = n + α & G d T < α < ( T C & _+ + α " P O& < n = k. T"& ^ 8$ T & $ T^&& =" 8 & B & = < α < T = α + V T((x + T )) = ((x))

d . < %! R @ & 5& y = f (x) + i: w 0 %&E ="& N x = b < x = a s < & %D =%! < U& ^ +

L ' P < @< mK! < &< L U& +P +&+& d ) y = x − [x]

((x)) = x − [x]

= (( + (T − α))) = (( − α + T )) = (( − α)) =

−α

=%! 5+DX C ' T mH =%! / α = 5 @;& r4 5+DX C ^ 8$ ' 8 & B & , =T = V+S + T%! ((x + n)) = x

) y = sin{π(x − [x])}

) y = ( x − [ x])

>0

d L ! mK! < @ h j: ^ L vP & & ^+Q< 3D ^ ="& U& G !+H @ 5Q; =%! w,G < + x &

7 Ok.5 TG +& & =V+ :5 δ T (Q < ε vP @5Q; & ^ =V+ v] L V GQ $ L @;& /& 5 7 < |x − x | < δ ⇒ |f (x) − | < ε

5!4

=%! " h L ,e V GQ S$

=

+ i : (x − ε; x + ε) @L& x L −ε ' L p L& @C. Tx L −ε Lp =%! R L

2

Tf (x) = x + x + i: ^ lim (x + x) V+ C T5 = = < x = x→ %! 5 & ∀ε∃δ∀x

< |x − | < δ ⇒ |x

V+ Lld |x |x + x −

+ x −

|

| &

+ x −

| < ε

=

|x + ||x − | < |x + |δ

< U& ' y = f (x) + i: / f @ d L ! 8.P−ε ' %! C < " [5 x L G.P ' f G +& & T =S. [5 x @;N Ok. + x & x /< y = f (x) V+hS 4 ' T (Q < ε P L& %! ∈ R C && @ ! G.P−δ x P L & S %: $ < δ G.P−ε & f (x) C Tx @;N L (x − δ; x + δ) − {x} G 2D5& ="& M75 L ( − ε; + ε)

x

` [a I & ` '8P

mH Tδ < V+ i: δ ^&& =|x − | < V |x − | < δ +Q< i: && $ =|x +x− | < δ + = < x+ < < x < < vX %! ^ vP δ fkD/ =δ < ε δ < ε & mH T"& ε O& < ^ _.P /& & T^&& =δ ≤ V& i: =δ = min , ε V+ i: %! : Ti: < δ O& c V+P

ε δ = min , V+ i: Tp ^ & =%! %! ε L =|x − | < < |x − | < mH =|x − | < δ < − < x − < S + |x − | < " ^&& T < x + <

` !K ^ ` '8P

∀ε∃δ∀x

`! b)-! ` '8P

|x + x −

| = |x + ||x − | < ×

ε

< 0 ' B <0

G.P−ε ' L %! 2DC C. ^ U/< ="& =V "& d L x @;N x L

`δ [a V ` '8P

|(x + )(x − )|

∈ R

(x − ε; x ) ∪ (x ; x + ε) = (x − ε; x + ε) − {x }

=

< ε

< ε, x

< |x − x | < δ ⇒ |f (x) − | < ε

. & < + x ;N y = f (x) U& 24 ^ =V+P x→x lim f (x)

&& + x & x /< f ” C ^ $ V8 ^ < T%! N; V8 ' 2DE & ,5 ”%! L & %! : mH TS& δ B< & 74 C

=ε

>>

:*1; $

6 # 5

%! " i:

=%! . P& <

√ x+ − − = x √ √ x+ − x+ + = ×√ − x x+ + (x + ) − = √ − x( x + + ) √ − x + = √ − = √ ( x + + ) x+ + √ √ − x+ x + + = √ ×√ ( x + + ) x+ + − (x + ) = √ ( x + + ) |x| δ √ = < √ ( x + + ) ( x + + )

f (x) = T^&& q lim

x→

x+ = − V+ %&E x−

2 N

V+ %&E V+P TG +& & = = − < x = T

x+ x−

x + ∀ε∃δ∀x < |x − | < δ ⇒ − (−) < ε x− x + =V+ Lld − (−) & δ x− x + x − x − + = x − = |x − | |x − | < |x − | δ

` [a V ` '8P

=δ

` [a I & ` '8P

V+ i: δ ^&& =|x − | < V T|x − | < δ i: && $ − M7; / L mH = < x − < − − < x − < mH Tδ ≤ V+ i: δ w-δ ≤ i: $x = < x − < V T^ :S ^&& =− < x < V T|x| < δ +Q< i: && $ x+ = x− + < / δ = δ + √ √ + = < x + < < x + < < vX %! ^ vP δ √ x+ − δ δ δ f k D/ =δ ≤ ε δ ≤ ε & mH T"& ε O& < − = √ < = x ( x + + ) ( + ) ^ _.P /& & T^&& =δ ≤ V& i: < vX %! ^ vP δ ε δ = min ,

V+

i: %! : Ti: < = fkD/ =δ ≤ ε δ ≤ ε & mH T"& ε O& < ε O& c V+P ^ _.P /& & T^&& =δ ≤ V& i: ε =δ = min{ , ε} V+ i: %! : Ti: < δ = min , V+ i: Tp ^ & =%! %! δ O& c V+P S + |x − | < " L =|x − | < δ < < δ = min{ , ε} V+ i: Tp ^ & =%! %! ^&& T < |x − | < " L =|x| < ε < |x| < Tx = mH = < |x| < δ x + < < x + < − < x < V+S + |x| < √ √ x − − (−) = |x − | |x − | V T @7 2D!3 M&; ]Q T < x + < δ≤ × ε =ε √ < x+ − / / |x| ε − = √ < =ε x ( x + + ) ( + ) =%! . P& < =%! . P& < ≤

` [a I & ` '8P

` !K ^ ` '8P

` !K ^ ` '8P

`! b)-! ` '8P

`! b)-! ` '8P

& %! 5& f : R → (; +∞) + i: = lim f (x) = + %&E = lim f (x) + x→

x→

:X L =f (x) + ' ε

>

f (x)

≥

mH Tf (x)

>

q lim

2 N f (x)

√

x→

x+ − x

TG +& & = =

=

$ =

2 N

V+ %&E V+P √ < x = Tf (x) = x + − T5 x V+ %&E V+P =

√ x+ − ∀ε∃δ∀x < |x − | < δ ⇒ x √ x+ − − @5Q; & δ x

= P L & T^&& T lim f (x) + f (x) x→

G d T|x| < δ S B< δ >

− <ε

` [a V ` '8P

≤ f (x) + f (x) − < ε

x =

>9

5 T < |x| i: & & B qV+ Lld

6 # 5

:*1; $

8" & x→−∞ lim f (x) = +∞

" + < ^ L

∀M ∃N ∀x x < −N ⇒ M < f (x)

≤

≤

8" & x→+∞ lim f (x) = −∞

< δ

mH T

x→

≤

G +& &

lim

x +

−x

x→

< |x − | < δ

V+ i:

x

δ M =δ = min

G +& & qx→+∞ lim

x

=−

) )

< ε + ε

< ε + ε ^&&

lim (x + ) =

x→

x− lim √ x−

=

x→

∀M ∃δ∀x < |x − x | < δ ⇒ M < f (x)

8" & x→x lim

5 ^ =

M

,√

M

δ " i:

V+ %&E

∀M ∃δ∀x

f (x) = −∞

< |x − x | < δ ⇒ f (x) < −M

8" & x→+∞ lim f (x) =

2 N

∀ε∃N ∀x N < x ⇒ |f (x) − | < ε

x ∀ε∃M ∀x x > M ⇒ − < ε

8" & x→−∞ lim f (x) =

x <ε =

f (x)

−

δ

≤

=

+

/ > = δ

< √

w0=> x

<ε

< < %! U& y = f (x) + i: T24 ^ ="& N+N C 8" & x→x lim f (x) = +∞

5 T"& < < S_& M L d 2DC %! LO %! : mH =δ

C < x d V 5Q; P T"& +∞ < −∞ && < x 8 & & = N+N d =%! [5 &/ L G %P =V+ %!: L [5

x+ x+ > (x − ) δ

w => x

<ε

!

]Q < V+ i: , = < δ (x − ) ^&& = < x + < + < − < x − <

mH Tδ

) lim (x − x) = −

)

x+ ∀M ∃δ∀x < |x − | < δ ⇒ >M (x − ) )

f (x) −

+ %&E L < L ' P

< (x −

f (x)

−

= De C ε < δ TN TM d TV+ [5

f (x) −

=%! " 2DE V8 < f (x) −

∀M ∃N ∀x x < −N ⇒ f (x) < −M

2

≤

8" & x→−∞ lim f (x) = −∞

V+ %&E

f (x)

−

&& < & 4 < ! < & w => x ^+:X S S + TV+ U.B w0=> x

∀M ∃N ∀x N < x ⇒ f (x) < −M

q lim x + = +∞ x→ (x − )

+ f (x) −

∀ε∃N ∀x x < −N ⇒ |f (x) − | < ε

4 $ V+!H C ^ 2DE & x

8" & x→+∞ lim f (x) = +∞

T%! Q<_ 5& y = log/ x $ = < ε T5 " i: %! : 8 5 =x > log/ ε V mH =M = log/ ε

∀M ∃N ∀x N < x ⇒ M < f (x)

>6

:*1; $

1 <# = >$ 7

sin(x) = x

) lim

x→

)

ex − = x x→ √ n +x− ) lim x x→

)

)

lim

)

x→∞

)

lim ax =

=

lim

ln(x + ) = x

lim

− cos x = x

x→

x→

x→

|a| < a=

x + x+

S S

ex − x x→ − cos x = sin

lim

x→

ln(u + ) = u u→

÷ lim

=

x + x+

+ Tu =

sin u u→ u

) lim

x→

)

lim

=

ln(x + ) = exp( ) = e x x→

wIx 2DE

u→

=%! . P& <

< 5& y = g(x) < y = f (x) + i: R 24 ^ a, x ∈ R )

) ) )

lim af (x) = a lim f (x)

x→x

x→x

lim {f (x) ± g(x)} = lim f (x) ± lim g(x)

x→x

x→x

lim

x→x

x→x

^ Tx→x lim

x+

>x−

+=x− =M

g(x) = y

x− x

lim

x→−∞

= −∞

)

x→−∞

x = +∞ − x

)

x→+∞

>N−

%! : mH

"

lim ( + x ) = lim

−

x √ x+

= −∞

sin x lim sin (x) sin (x) ÷ x x→ x lim = = = lim x→ sin(x ) x→ sin(x ) ÷ x sin(x ) lim x→ x

x→x

f (x) = lim f (x) ÷ lim g(x) x→x g(x) x→x

lim f (g(x)) = 24

x→x

x→x

lim f (x)g(x) = lim f (x) × lim g(x)

x→x

+

75 8 ,< =%! {+C < ε − σ +S \< +& 8 d < < " L c L N %+ \< L ! T8 < ^ Q4 =%! +G /< k5& \< ^ ="& ym85Q& < & Ǳ_B L % z $ & %! {+ ^ & T" P ! 25: & _+ / A/ mK! TS 2DE yHz y74 z n; P Y+ ! U& { & +t+H U& < 7 P 5 & 7 P Td %! < " = TB A/ < " < L ! & < W+H @7 V+ & %! : \< ^ L ! & Tfke =%! " ! ! < L 4 $ & sin x V+ & S ,O ! G d T%! ' && lim x→ x

x

2

x+

0 9!: ;# 6

lim ( + /x) = lim ( + u)/u = e

x→∞

lim

/x V+ i:

+

+ %&E

x→

=x−

=N = M + N −

V TX = eln X 8 & B & wFx 2DE x→

=x−

=

V+ B Tw9x 2DE + < u = x V+ i: ^&&

lim ( + x)/x = lim exp

>M

x+

x

sin u − cos x = lim = u→ x u

+ x > N +Q< i: L mH TN ≥ %! ^ & i: ^&& T > x > "

`b)-!

k5& < V]H 1 < 0 T Q: %34 fk5: = V+P 2DE ^ y, +HP @C/z %./ + =u = ex − V+ i: w>x 2DE ^&& < x = ln(u + )) lim

N;

V+ B & C ^ 2DE &

lim ( + /x) = e

2 N

V+ %&E

x + ∀M ∃N ∀x x > N ⇒ x+

x

= +∞

) lim ( + x)/x = e x→

n

x + x+

+& & qx→∞ lim

2+

S =x→y lim

/& L s&<

f (x)

>F

=

& 43 4K

1 <# = >$ 7

:*1; $

`b)-!

! ! M+N3 < . 2DE w>x =VK! & L & =x→x lim g(x) = m < lim f (x) = V+ i: x→x

! O& 8" & N U& <_Q " B U& d G d T" ! O& \< & 5& S 87& T"& " =%! N x + N f (x) = U& $ x − / f @ x = L G.P−ε ' $ < %! Lx ^&& TwDf = R − x + lim = f( ) = = x→ x −

2

< ε =

ε =

x + + mH T%! N 5& x + $ x+ '
lim

x

x→−

(− ) +

=

(−

+ + x

V f (x) = tan x − sin x U& x

V+ i: (Q ε >

ε + max{| − |, | + |})

|f (x)g(x) − m| = |f (x)g(x) − f (x)m + f (x)m − m|

=

) + + (− )

(

G d |x − x | < δ S B< δ 24 ^

G d |x − x | < δ S B< δ < |f (x) − | < ε

G d |x− x | < δ S B< δ < |g(x)− m| < ε

G d Tδ = min{δ , δ , δ } S mH =|f (x) − | <

2 N

ε ( + ||)

2 N

≤

|f (x)||g(x) − m| + |m||f (x) − |

≤

max{| − |, | + |}ε + |m|ε

<

max{| − |, | + |} +|m|

tan x − sin x sin x cos x − lim = lim x→ x→ x x x sin x − cos x = lim lim lim x→ x x→ cos x x→ x

(

(

ε + max{| − |, | + |})

ε <ε + |m|)

×

=

Ty

=

×

ex − x→ sin x

lim

x→e

=

x i: & Tf (x)

x−e e

=

ex − sin x

U&

2 N

V

T7.B $ T< B T; T%&E U& + i: S T Y+34 T%De &x TQQ]P Te7e U&z M7; / < . GQ T. Tw S < %De ' ,.C + i: ^+t.P qV+& yL TVP L U& < M TVP & U& < U.B TU& ' C D!3 TVP & U& < {+ TVP & U& < V+N TVP U& < wU& @ 8$ T5x U& ' 3 TU& ' << yL ,.Cz V+ !& G U& & 5& O& < 24 ^ =V+&

= lim

i: & Tf (x) =

ln x − x−e

= lim

y→

= lim

y→

=

sin x ex − ÷ x x x→ x e − sin x = lim ÷ lim x x→ x→ x y e − = lim ÷ = ( ) ÷ = y y→

lim

y=

@ & y = f (x) U& 2 4 FG ' x ∈ D P ǱL & V+hS N 4 y = f (x) U& " %: D d L G.P−ε EF . & N U& @.P @C. ="& f (x ) && x =V+P W.

D

e

ey

lim

y→

lim ( + y)/y

y→

=%!

e

ln x − x−e

U&

2+ N

V

ln(ey + e) − ey

% "*+ ," -%" *+ ./ %0 ,1 2"/+ ," % "*+ 3 4 ,1 ' %0 5/6 7/8 y = f (x) 2+ 9" .*: ; " + <= . >$*% f " x ," <:?−ε 3: 4 @ " f (x) " 4 // lim f (x)

(ln e + ln( + y) − ) y

ln( + y) =

e

lim ln ( + y)/y

y→

< %! N 5& ln e

ln e &&

( + y)/y

$

x→x

N ^&& T"& e &&

=%! D/ @+A/ L V+N + +A/ ^ >I

?@4 37

:*1; $

<=3 2"6

mH T N

' < "& N y = f (x) S V D/ W(& U& G d T"& " / d @ x L G.P−ε ! G.P−ε ' U/ L & =%! f (x ) && x y = f (x) 5 T / y = f (x) @ x L U& B< Q< T". [5 3& @;N x = −

lim

x→−

√ < %! N f (x) = x + x+

@;N \

lim

<

2

=

)

=>=> @+A/ L ! & < < x lim

x→

√

−

x

−

√

−

x

= y

= lim

y→

i: &

− y

−

y +

2% N

( &

y

y

×

lim

y→∞

+

y

x→

lim

x+ x +

+

x→

lim

x→∞

+ x

+ x/

) lim sin (πx ) x→− ) lim

x→

) )

lim

x→

lim

x→

−y

>1

+ −

) ) )

tan x − sin x

)

cosh x −

)

sin x

sin x

+x

x +x+ ) lim x x→

tan x x

αx βx ) lim e − e x x→ √

) lim x + x→− x +

√ ) lim x − + x x→ x +

V

!

√ x + − x

lim arctanh

)

2 N

=

lim ln(cos (ln (x + x + )))

)

2 N

=

+

y→∞

x→

)

−x+

−/

y+

= lim

lim

lim

)

(x + )(x − ) = lim x→ (x + )(x − ) x+ + ? = lim = = + x→ x + √ −x− ' √ U& < x+ −x+ ^&& T && x = − L ! G.P−ε √ √ √ −x− −x− −x+ lim = lim ×√ x+ x+ x→− x→− −x+ ( − x) −

√ = lim x→− (x + )( − x + ) lim √

x

y→∞

x→−

x − lim x→ x − x −

x→−

− −

=

+ D!3 L < L ' P N

x+

?

x)/( +x)

= e × = e

^&& T && x = _& B .P g(x) = x +

=

(+

x+

x

U&

+

x→∞

%: $ < ε C S . SG F>) < |x − x | < ε < r4 x P ǱL & " B< C < B< G d T"& /& f (x) = g(x) < x y = g(x) B< C < B< & x y = f (x) =x→x lim f (x) = lim g(x)
x − x +

V Ty =

G S. [5 f (− ) Q T%! && V+ 5 y = g(x) G U& & y = f (x) U& T L ! G.P−ε ' y = g(x) < y = f (x) X =S [5 _+ x y = g(x)

2# N

+ x x − < U& $ x + + x (+ x)/( +x) x−

) )

lim x→

−

x + x − x − x +

lim (x − x + )

x→

lim

x→

cos x − cos(x) x

sinh x x x→ lim

lim ( + x)/x

x→

lim

x→∞

x − x +

ln(cos x) x→ x lim

x

?@4 37

:*1; $

)

√ x+ lim √ x+ x→−

)

)

arccos( − x) √ x x→+

)

/x cos x ) lim x→ cos(x) √

) lim x cos x

= lim

y→

x→

( − y)(−y + y + ) y→ ( − y)( + y + y )( − y)( + y)

−y + y + ( + y + y )( − y)( + y) ( − y)(y + ) = lim y→ ( + y + y )( − y)( + y) y + ? = lim = = × y→ ( + y + y )( + y) ?

= lim

y→

lim

lim

x + + x

+ x + x −

C S T S5G )& & 43 . SG F>) x ≤ M P ǱL & " %: X < M G d T"& B g U& %+& < f (x) = g(x) =%! && 7D/ & < B< _+ f %+&

x→

x √ √ x + − x + √ ) lim √ x→ − x − x − # $ x x lim − ) x→+∞ x − x +

) lim x − + x x→−∞

)

lim x

x→+∞

)

lim

x→∞

x +

x x

+ +

V y =

)

lim

x→

+ tan x + sin x

/ sin x

x

) lim √

+ x − ( + x) + αx − n + βx ) lim x x→ x /x a + b x + cx ) lim x→

/x i:

(x + a)(x + b) − x (a + b)x + ab = lim = lim x→∞ x→∞ (x + a)(x + b) + x (x + a)(x + b) + x

V y =

cos(xex ) − cos(xe−x ) ) lim ln(cosh x) x→ x→ ln(cos x) x $ # √ √ ) lim x+ x+ x− x

y→

y

+ y + y +

= lim y→ y +y+

√ √ √ x)( − x) · · · ( − n x) x→ ( − x)n− ) lim { n (x + a )(x + a ) · · · (x + an ) − x} lim

− y + y

+

y −y+

) lim

x→

=

+

=

( −

%!& L < L ' P N

x→+∞

2 N

&

= lim

lim

x→+∞

)

/x i:

+x+x − −x+x = x→∞

= lim +x+x − −x+x x→∞ +x+x + −x+x × +x+x + −x+x x = lim x→∞ +x+x + −x+x lim

/x x + x + ) x→∞ lim x − x + √ √ ln( + x + x) √ √ ) lim x + x) x→ ln( +

)

a+b+ a+b = lim √ = y→ + +

a + b + aby = lim y→ ( + ay)( + by) +

√

(x + a)(x + b) − x = x→∞ (x + a)(x + b) + x (x + a)(x + b) − x = lim x→∞ (x + a)(x + b) + x

m

x→

2

&

lim

−x

x +

+& L 2j& & U: @+A/ T%+& P S

x+

)

+ y − y ( − y )( − y )

= lim

ln( + x ) x→−∞ ln( + x ) −x ) x→∞ lim arcsin +x

) lim (x + ex )/x lim

lim

)

x→+

x→−

x→

+x +x

ln(tan x) − cot x x→π/ sin x − π ) lim − cos x x→π/

lim

)

lim

+ xe−/x sin

e/x

x + x + x→ x + x +

) lim

x

>J

)

√ lim x→

:*1; $

^ & q

lim

x→−

#(A $

2 N

V+ %&E " %&E & p

= + e/x

< − x < δ ⇒

∀ε ∃δ ∀x :

+ e/x

)

lim

x→

+e−/x arctan

f (x) =

+ e/x

−

−

δ

+ e/x

=

= ε

=δ = −

D!3 L @:;8 < L ' P x − |x − | − cos(x) ) lim x x→− ) lim arctan −x x→−

)

)

)

)

) )

lim

x→−

lim

+ e/x lim x |cos ( /x)|

x→+

[; ∞) d @ < %! N 5& f (x) =

x→+

sin

√

x V+

' ;N P y = f (x) + =%! =%! && f (x ) & < B< %! [; ∞) G.P−ε S 5 %+ [; ∞) G.P−ε ?+P x = . d - & $ ^ S. [5 Tx < - V+.5 [5

ln ε

+

x − |x − | − cos(x) ) lim x x→+ ) lim arctan −x x→+ lim

x→+

< %! U& ' y = f (x) + i: Tx < x < x + ε P ǱL & B< $ ε De C (x ; x +ε) ⊆ %P ε T5 TS [5 x f (x) && x = x y = f (x) %! V+hS 4 =Df x→x lim f (x) = l V+ < %! l

lim x [ /x]

x→−

< x − x < δ ⇒ |f (x) − l| < ε

∀ε ∃δ ∀x :

x→−

lim

S S S

"!&> #

V+ i: %! : mH

2

2

$

e/x e/x < < e−/δ + + e/x

5 Te−/δ

= ln ε

n>m n=m n<m

`b)-!

/x <

x

an xn + · · · + a x + a bm xm + · · · + b x + b

⎧ ⎨ ∞ an /bm lim f (x) = x→∞ ⎩

<ε

− S + −δ < x < i: L δ /x < e < e−/δ ^&& T%! 54 5& y = ex

=

e/x

24 ^ T"& 4 [Q(

−

x

+xe−/x sin

U& bm < an { S +P

− < ε

G +& & T < ∀ε ∃δ ∀x : −δ < x < ⇒

x √ √ x + x sin x

=w^.x [5 x→x lim f (x) = l & 2j& − ^(! _+ x→x lim f (x) = ∞ $ < L ^+t.P + =w^.x %S

< Q.5 < & :;8 < D & L @+A/ L =S ! 4 Q.5 mK!

q

2

V+ %&E V+ %&E & p ^ &

lim x [ /x] =

x→+

2

∀ε ∃δ ∀x :

2

r4 x P ǱL & " %: $ ε > S < T"& /& f (x) = g(x) < T < x − x < ε < B< _+ x→x lim f (x) G d T"& B lim g(x) S _+ x→x − =%! && 7D/ & <

< x

< x − < δ ⇒ x

i: & + Tx −

x

−

≤ [x] < x

−

<x

x

" i: %! : 5 =|x [ 2

9)

V+

−x = x

− < ε x

ǱL & " %: $ ε > S f (x) = g(x) < T < x − x < ε < r4 x P lim f (x) G d T"& B lim g(x) S _+ < T"& /& x→x + x→x =%! && 7D/ & < B< _+

−

≤x

x

−

`b)-!

V

=

/x] − | < |x|

^&& =δ = ε

B$81 CD '

_+ x = x y

:*1; $

G d Tx→x lim g(x) = lim h(x) x→x =%! && d } N & < B<

8 & : < LO " U- lim f (x) < lim f (x) %! ^ "& B && < B x→x x→x + ="&

= f (x)

lim f (x)

x→x

ǱL & %P M + i: S T24 ^ =g(x) ≤ f (x) ≤ h(x) x > M P lim g(x) = lim h(x) B< _+ +∞ y = f (x) G d Tx→∞ x→∞ =%! && d } N & <

G d T

f (x) =

lim x

x→

<

<

+ + ···+

24 ^ T

|x|

x→−

<

x x

.

+ + ···+ |x|

.

|x|

=

+

=

|x|

|x|

−

x .

≤

|x|

<

|x|

.

)

)

|x|

x .

|x|

−

| sin x| x x→+ lim

)

lim ( + |x|)/x

)

x→+

"

| sin x| x x→− lim

lim ( + |x|)/x

x→−

⎧ x < S ⎨ x − x + ) f (x) = ⎩ (x + )/(x + ) ≤ x ≤ S sin (π/x) < x S ⎧ x ≤ S ⎨ ) f (x) = ⎩ e/x <x< S x ( + /x) ≤ x S √ √ ) f (x) = [x ] ) f (x) = x − [ x]

+

)

f (x) = (− )[x

]

)

f (x) = sgn(sin x)

.(|x| + )

& 24 & ≥

x→

XN . " U& L ' P %! < $ + D!3 %! . ++a U& @;&

|x|

x→

+ D!3 L @:;8 <

A'

S S

lim f (x) = lim (−x + ) = − + =

−x + x ≤ x − x >

2 N

lim f (x) = lim (x − ) = − = −

x→+

2 N

V+ i:

< |x| <

^&& < =

=

|x|

+ i:

<x<

24 ^

x P ǱL & $ 1 ^&& < − ≤ sin ( /x) ≤ V < x ǱL & mH T y = −x < y = x U& =−x ≤ x sin ( /x) ≤ x ^&& T%! 4 && + 4 & %! L x /< S %&E & 24 & = lim x sin ( /x) = x→+ = lim x sin ( /x) = + = lim x sin ( /x) =
^&& =

x→

≤ sin x ≤

x→−

!

lim [x] sin(πx) = lim sin(πx) = sin π =

+& B< _+ :;8 < & & A/ =VK! & ^. C & d 24

x→

2

S

<

x→+

P ǱL & %P M + i: S T24 ^ =g(x) ≤ f (x) ≤ h(x) V x < −M _+ −∞ y = f (x) G d Tx→−∞ lim g(x) = lim h(x) x→−∞ =%! && d } N & < B<

−

< x−

.

|x|

=%! " %&E V8 <

= lim

x→

?# 70 @A

.( − |x|)

.( ± |x|) =

&/ Q.5 P\< & < 2DE & +A/ ^ L 8 < %! 7 < d \< < + P& @ !@ = ! T%! OX L n" & 8" ! +A/ ^ =%! <3 +& d "&

2 N

lim (e − )/x

=+ D!3 x→∞ N =ex ≤ (ex − ) ^&& < ≤ ex G d Tx > S = e ≤ /x (ex − )/x + x

e ≤ (ex − )/x < (ex )/x = e /x

=%! e && p < x→∞ lim

e

/x

=e

%

& %P ε > + i: S =g(x) ≤ f (x) ≤ h(x) V T|x − x | < ε & x P ǱL

9

=$ ? E F ,

:*1; $

24

+ D!3 L < L ' P

V+ i: =V+P / || && ε q = V+ i: r4 |x − x | < δ " x S C < %! (Q δ > ^&& < Dri(x) = 24 ^ =

) lim x |cos ( /x)|

)

|Dri(x) − | = | − l| = || ≥ ε

)

=%! . P& {+ ^ & <

x∈ /Z x∈Z

S S

8 & : < LO " ! L _+ < %! L x y = f (x) %! ^ "& B ="& && < B $ lim f (x)

x→x

= x − [x]

U& V+ %&E

lim f (x) = lim {x − } =

x→+

x→+

lim f (x) = lim {x − } =

x→−

x→−

−

(k ∈ N)

*

y = f (x) U& 8 & LO " ! ' y = f (x) %! ^ T"& " x = x ∈ R P ǱL & 8 < [5 x L G.P δ > P ǱL & " %:& $ ε > C ' < < |x − x | < δ "& " B< X x ∈ R ' G +& & =|f (x) − | ≥ ε

!

=

k + + ···+ , lim x. x x x x→+

d ="& Zj(& +A/ T\<

=%! " & fkD/ 6=9=>+

f (x)

lim x ln x

x→+

:# < BC

/: Z − {} @C. N @.P L U& + %&E w0 %! cot (πx)

)

P ([x])

/: x = N . f (x) = xDri(x) + %&E w ="& 4 && < B x = d < %!

f (x) =

lim ( /x) sin ( /x)

x→+∞

lim x [ /x]

x→

=%! %De { & 7.B $ ' P (x) + i: wF [P (x)] = +P lim =x→∞

!

)

x→

x=

= ,

− = lim f (x) x→+

= B< x = y = f (x) ^&&

∀ ∃ε ∀δ ∃x :

< |x − x | < δ <

|f (x) − | ≥ ε

! !

N f (x) = =

78 U& V+ %&E

= sgn(x − x + )

U& + %&E w = x = < x =

Dri(x) =

A sgn(sin(x)) U& +P w0 πZ = {nπ | n ∈ Z} @C. N T5

π

x∈Q x∈ /Q

!

S S

=%! /: R N @+7 %Q < =x ∈ R V+ i: Tp ^ & ="& S x w < "& S x w[Q d W+H %! ^8. %Q ' ]Q < P VP & +D" +& < ^ Tx ∈ Q V+ i: ^&& =V+P / 3& =Dri(x ) = + V+ i: =V+S ' && ε q = V+ i: r4 |x − x | < δ " x S C < %! (Q δ > ^&& < Dri(x) = 24 ^ =

< T. 5Q; PQD j: L mH 5& %./ = d @5Q; L k /

23(!

U& 8 & : < LO " " ! ǱL & %! ^ T"& && x = x @;N y = f (x) G.P l & {f (xn )}∞ @ QD Tx & G.P {xn }∞ @ QD P n= n= ="&

|Dri(x) − | = | − | =

90

≥

=ε

:*1; 2

:*1; $

%! !+H %! L x = x @;N y = f (x) w ="& f (x ) && x f %! $ %! !+H $ L x = x @;N y = f (x) w| ="& f (x ) && x f

2

U& V+P ! PQD p ^ & = x = 24 ^ =V+S p zn = nπ+π/

y = sin ( /x)

< xn

= nπ

lim f (xn ) lim f (zn )

S + %Q! & T^&& y = f (x) U& 8 & : < LO "

"

f (x) = sin (

"

_+ < %! L U& ^ %! ^ T"& !+H x = x @;N ="& !+H x = x $ L

/x) %+ ^8. T^&& = = L T"&

S N L U& V+P f (x) =

y = f (x) + i: " "U: G !+H ' x = x 4 =%! !+H & 2j& ="& B x f V+hS y = f (x) $ L "U: G !+H _+ < %! L "U: G !+H G !+H %! < $ L x f S =%! [5 &/ "U: x f G !+H <_Q T"& "U: && Q< T"& B x f %! < $ T5x ="& w="D x = x

<

n /(n + ) (m, n) =

2 N

x = m/n ∈ Q x∈ /Q

S S

S C L QD {xn } V+ i: =x ∈/ Q V+ i: a " ! 2j& xn = n < %! G.P x & "& bn 24 ^ ="& " lim f (xn ) = lim

n n+

= +∞

T + S C ' & S C L QD S L + %+& & QD ^ @ L! P |( < 24 k_Q =

y = g(x) < y = f (x) U& S " < "& !+H y = f (x ) y = h(x) < "& !+H x = x Tf (x) − g(x) Tf (x) + g(x) Taf (x) U& 24 ^ Ta ∈ R S = !+H x = x f (x)g(x) < h(f (x)) Tf (x)g(x) =%! !+H x = x f (x) ÷ g(x) G d Tg(x ) =

f (x) = [

/x]

U& +P

2

!

= x = %! /: S N . L U& +P ∗

f (x) =

%! N 5& y = f (x) + i: " 24 ^ =x ∈ Df < w" B >=0=> &x

2

sinm (xπ)

(m, n) =

<

x = m/n ∈ Q x∈ /Q

D(0 E

G d T[x ; x + ε) ⊆ Df S %: ε > S w =%! %! !+H x = x y = f (x)

S S

1

d V+! & T5 -%! j U& . 4 $ < V& ]l < L & $ V7/ B< <_Q 7 ^ & !H W(& ^ -V+P / d & k B< G !+H ! =%! w| q$ L G !+H w qw! < Lx G !+H w[Q =%! L G !+H

G d T(x − ε; x ] ⊆ Df S %: ε > S w0 =%! $ !+H x = x y = f (x) T(x − ε; x + ε) ⊆ Df S %: ε > S w> =%! !+H x = x y = f (x) G d √

x =

= lim sin(nπ) = , π = lim sin πn + = .

2

[5 x f < y = f (x) + i: " U& V+hS 4 =" x f %! !+H x = x @;N y = f (x) w[Q ="& f (x ) &&

=+G& p f (x) = x U& ! " [; +∞) & && d @ < %! N y = f (x) $ !+H x ∈ (; +∞) N . y = f (x) mH T"& ="& %! @ !+H x = @;N < %! 9>

:*1; $

:*1; 2

2+ N

y = f (x) $

2 N

$ =+G& p f (x) = arcsin( − x) U& & && d @ < %! N y = f (x)

=+G& p f (x) = /x U& y = mH T"& R − {} & && d @ < %! N $ T
Df

=

2# N

≤

lim xDri(x) x x∈Q = lim x ∈/ Q x→ x→

lim x =

≤

2 N

U& $ =+G& p f (x) = sin x U& x T"& R − {} & && d @ < %! N y = f (x) L =%! !+H x ∈ R − {} N . y = f (x) mH =%! ' & && < B x = x y = f (x) T:X ="& "U: x = @;N y = f (x) G !+H mH Tf () = 2j& x = f U& [5 & T5 =d %!& x = x !+H 5&

S S

x→

U& ^ T5 =%! !+H x = y = f (x) + ="& !+H x =

)

y = arcsin(x )

)

y=

y = x [ /x]

) y = tanh

x

−x

) y = ((x)) = x − [x] y=

)

y y

)

y

)

y

)

y

)

y=

)

√ x

lim f (x)

x→+

y = ex+/x

)

= =

y = sin(cos (tan x))

lim f (x)

x→−

)

= =

y = sgn(sin x) − cos πx ) y = −x

cos ( /x) cos ( /x) + = arctan x x−

π = sgn sin x x x x ∈ Q S = x ∈/ Q S cos (πx/) |x| ≤ = |x − | |x| > sin(πx) x ∈ Q S = x∈ / Q S

)

)

y=

x +x ) y = cot π x

sin x |x|

)

)

2 N

|x| ≤ S U& p f (x) = xx + |x| > S ' y = f (x) G d T|x | < S 24 ^ =+G& y = x U& Q =%! x & && x L G.P−ε @;N y = f (x) ^&& q"& R d @ < %! N =%! !+H x ^ =x = − 8 < x = G d T|x | = S _+ < %! $ @ !+H x = @;N y = f (x) 24 L T"& %! L "U: G !+H

< G !+H TL U& L ' P " " + !& ^8. N . " U& G !+H C ) y=x

x − ≤ −x≤ x ≤ x ≤ = [; ]

T%! !+H x ∈ (; ) N . y = f (x) mH T"& @ !+H x = @;N < "& $ @ !+H x = @;N =%! %!

&x +G& p f (x) = xDri(x) U& U& ^ =w" B 0=F=> & Dri 78 U& [5 @p f =w%! < $ x %! /: x = N . U& ^ T =%! !+H x = N . T^&& L T%! f () = && x =

=

lim (x + )

x→

=

= f( )

lim (x )

x→

= f( )

_+ < %! @ !+H x = @;N y = f (x) T^+t.P L T"& $ L "U: G !+H lim f (x)

x→−+

= =

lim f (x)

x→−−

= =

S S

lim (x )

x→−

= f( ) lim (x + )

x→−

=

= f( )

&& x L G.P−ε ' y = f (x) G d T|x | > S @ < %! N y = x+ U& Q =%! x+ & =%! !+H x @;N y = f (x) ^&& q"& R d 99

:*1; 2

:*1; $

2 N

= ' /O : @B & 7.B $ P < an P (x) = an xn + · · ·+ a x + a " i: S TL x /< P (x) T24 ^ =w%! & an < %Q x b > ' /O mH T%! +∞ && + +∞ & + −∞ & x /< P (x) T:X L =P (b) > %P %P a < ' /O mH T%! −∞ && 5& P (x) T =P (a)P (b) < + =P (a) < T^&& =%! !+H [a; b] & ]Q < %! R & N d ǱL & B< x ∈ (a; b) T )=I=> @+A/ M&; =%! 4 P (x)

)

y=

x − / |x| ≤ /x |x| >

P = ". [5 x = L U& & mK! T%! "U: x " U& G !+H +P + U: d G !+H Tf (x ) {! [5 ) f (x) = ( + x)/x ,

)

f (x) =

x

e−/x ,

⎧ ⎨ − sin x f (x) = A sin x + B ⎩ cos x

⎩

− ≤x< x= <x≤

x−

S S S

U& + %&E w0 f (x) =

− ≤x≤ <x≤

x+ −x

x ≤ −π/ −π/ < x < π/ π/ ≤ x

U& +P w> f (x) =

(x + )

(−/|x|+/x)

x = x=

@L& & y = f (x) U& S 2- )4 # "

G d T"& f (b) < f (a) ^+& C c < & !+H [a; b] @ & =f (x ) = c B< b < a ^+& x

S S

T5x %! + N %+4 [− ; ] @L& & _+ < d N @.P G d T + N < S =%+ !+H [−; ] & Q Tw P +

x

FG

./5

( ) ;K * ) "

<

< & !+H [a; b] @ & @L& & =f (x ) = B< (a; b)

./ H'

&/K

"

T"& !+H [a; b] @ & @L& & y = f (x) U& S /J ǱL & " %: S & x , x ∈ [a; b] C G d =f (x ) ≤ f (x) ≤ f (x ) V x ∈ [a; b] P

|
[−π/; π/] @L&

2

& y = sin x U& " %! ]Hc85 T^&& =%! !+H < 54 k+ @ & @L& & _+ wy = arcsin x T5x d c85 < =%! !+H sin ([−π/; π/]) = [− ; ]

U& +P wF x≤− <x

U& S ∈ G d Tf (a)f (b)

y = f (x)

N . < %! 8 [a; b] @L& & y = f (x) + i: w9 y = f (x) + %&E = + f (b) < f (a) ^+& =%! !+H [a; b] @L& &

√ − −x √ f (x) = − x−

S S S

U& S /J FG TV+ Ǳ G "

G d Tf (x ) = < "& !+H x = x @;N y = f (x)

@L& & y = f (x) U& %fC " %: ε > = . ++a (x − ε; x + ε)

S S

Q< T%! !+Hx = _& [− ; ] @L& . & ="& L& d & .++ < V._ t.P

f (x) =

< & 8 k+ y = f (x) U& S " y = f (x ) @;N x = f − (y) G d T"& !+H x @;N =%! !+H

< Tx = @;N _& %! !+H [− ; ] @L& . & =.++ < V._ [− ; ] @L& &

)

< (Q N+N C xn < = = = Tx Tx + i: w0F ^ + _& ,e R @ & 5& T"& 2< ="& !+H N

U& + %&E w f (x) =

√ x|x|x , ) f (x) = sin x sin . x ∗

"& !+H R & L U& + ^++5 X B < A w06

"

⎧ ⎨ x+

S S

S S 96

:*1; $

@A&G H@%4 I

! & < V+N & %D IF @C. T^&& @N & +5 TǱ+" ^ @5Q; p & %+ & %! 2DC ( ^ & ^ =V+ n; d mK! < xa @! '$ %+& U& L ( L = ^ ǱAC & G P'$ %+& @N

Q< %! ]Hc85 < !+H [−; ] − [− ; ] @L& & =%+ !+H d c85

=>F G=$ 3

< & U: & U! k D "< h W(& ^ L vP & V" ] _+$ P L D/ =%! wVDx 8" & f (x + a) f (x) x→a lim @5Q; = 5Q; lim g(x) x→ g(x + a) ^ < x = a V+ i: B .P +Q ^+.P & =P . 3& %+7 L

%+& g < f f g VD %Q TW&& ! %Q & ln < exp Z L ! & T P '$

. D

f = exp ln f g = exp g. ln f = exp g

g

ln f

McC

4 y = f (x) U& <x lim f (x) = V+hS wx = a T <x x→ ="& _+ :;8 p qwx→a lim f (x) = T . & wx = a T <x '$ %+& U& @.P @C. T" u O& 7+Q & & =V+P wIFa T <x IF =%! a = V+ i: .P mH ^ L

g(x) x→ / ln(f (x))

%! : lim f (x)g(x) & mH x→ =V+ !& e . & 4 < . D!3

lim

%+& ' y = f (x) + i: " X c 4 [Q( C < k %De C S =%! '$

f (x) U& V+hS T24 ^ = lim k = c " %: x→ x k @D L < ]H N '$ %+& ' y = f (x) %+& @.P @C. =O(f ) = k V+ j & < %! ^ L =V+P W. IF . & ]H N P'$

=%H V+P IF ǱAC @5Q; & mH

2

lim

x→

=

− cos x

U&

− cos x = x

TVLK&

y = x sin ( /x)

U&

x sin ( /x) xk x→

2 N

2 N

& +A/ ^ " C @+7 TW&& & y = f (g(x)) _& T%! Y+34 (Q a =%! %! a = %Q < y = f (x) L (! O& @+A/ g(x) %Q %! ^ d +Q %! + y = f (x)g(x) ,e C & ="D '$ %+& U& ^ %! ^8. 7

sin x U& T8$ %+& sin x < x U& Q < x = + '$ %+& ]Q < %! ' && 4 (sin x)x ="& " 4 U& ^ %! ^8. =%! /: x < x sin ( /x) '$ %+& < %D Tfke

x sin ( /x) sin ( /x) = lim k x x→ x→ xk− lim

=w^.x B< k!!

G d Tk = S w lim

x→

− cos x Tsin x Tx

%+& y = g(x) < y = f (x) + i: ^ =%! 5& y = h(x) < a ∈ R T"& '$

Ty = f (x) − g(x) Ty = f (x) + g(x) Ty = af (x) U& T24 y = f (x)h(x) < y = f − (x) Ty = f (g(x)) Ty = f (x)g(x) =8$ %+&

@5Q; & S L T%! N &/ P %! ^8. %Q ! G d

G d Tk > S w[Q

lim

<

V 0=0=> @+A/ j:f& C &

L O( − cos x) = T5 T%! =%! < && |( x < "& 4 & [Q(

+l '$ %+& '

=E

Tx U& "& IF ǱAC T5 =8$ %+& U& L P . ="& 4 & && x = d L ' P ^&& < = + '$ %+& /x < cos x Q ln( + x)

'$ %+& ' y = sin x U& sin x L O(sin x) = T5 T%! ' @D ' && lim x→ x =%! ' && |( x < %! < @D '$ %+& ' y

H

x sin ( /x) = lim sin ( /x) xk x→

9F

@A&G H@%4 I

y = g(x)

^ =

<

:*1; $

=w^.x B< _+ ^

G d Tk < S w|

+ i: R < "& N &/ 8$ %+& 24

y = f (x)

= a ∈

)

) O(af ) = O(f )

O(f g) = O(f ) + O(g)

)

O(f ◦ g) = O(f ) . O(g)

)

O(f + g) ≥ min {O(f ), O(g)}

)

O(f (ax)) = O(f (x))

y = f (x)/g(x)

x sin ( /x) = lim x−k sin ( /x) = xk x→ x→ lim

)

O (f /g) = O(f ) − O(g)

'$ %+& T^&& =wIF ⊂ IF T^&&x "&

N &/ +l

y = x sin ( /x)

U& < y = g(x) < y = f (x) + i:

%! 5 & /< w9x @;& T DQ ="& '$ %+&

0=1=> @+A/ L w6x %./ $ < y = f (x) @D < L y = f (x) + g(x) @D @D!3 %! ^ !H -%! " ! < L y = g(x) T3& " ^"< & =%+ /& < 7 %Q + u L Qe V TO(−x) = < O(sin x) = Q w %! ^8. T^&& =O(sin x − x) =

O(g) G d Tc = S w[Q < qO(f ) = O(g) G d Tc = , ∞ S w =O(f ) < O(g) G d Tc = ∞ S w| m85Q& qc = G d TO(g) < O(f ) S w[Q < qc = , ∞ G d TO(f ) = O(g) S w =& P %+& c G d TO(f ) < O(g) S w|

8 & B& V+S + PD @N & TO( O(sin x)

O(f + g) = min {O(f ), O(g)}

−cos x

lim

x→

sin x

sin x

− cos x

= lim

x→

sin x

O(f + g) = O(f )

V TO(x)

#

=

'$ %+& ' ] +

V TO(x) = O(sin x) = Q w0 %! ^8. T^&& =O(sin x + x) =

f (x) =c = lim x→ g(x)

− cos x) = < − cos x = = lim sin x x→

8 & B & =%! −cos x

= lim

x

x→

=O

( sinx x )

− cos x sin x

=

= , ∞

^&&

=

5! < @D TL @+A/ < ,<B '. & =. D!3

< O(sin x − x) = Q w> %! ^8. T^&& =O(sin x − x + x) =

=

6; 4K

O(f ) > O(f + g)

+ D!3 y @D

2

=O(xa ) = a G d Ta > S

y = (sin x ) ln( + sin ( − cos x))

& %! && O(y) 24 ^ =

= = =

) O(

) O(sin x − x) =

) O(tan x) =

) O(ln(

) O(ex −

+ x)) =

=O(

O(sin x ) + O(ln( + sin ( − cos x))) = =

) O(sin x) =

√ n

+x− )=

) O(sinh x) =

O(sin x ) + O(ln( + x)) × O(sin ( − cos x))

=O (loga (

+ x)) =

=O(ax − 9I

− cos x) =

)=

G d Tn ∈ N S

) O(

) O(sinh x − x) =

O(sin x) × O(x ) + × O(sin( − cos x)) × × + O(sin x) × O( − cos x) +× ×=

2

2*

− cosh x) =

) O(tanh x) =

G d T

= a >

S

)=

G d Ta >

2 S 2

:*1; $

) (ax −

) (

n

@A&G H@%4 I

) (ex −

) ∼ x ln a x +x− ) ∼ n

) ( − cosh x) ∼ −

)

x

√ sin( x) ln( + x) √ x→ (arctan( x))

) ∼ x

lim

=%! 4 L 24 @D V+P %! : p ^ & = =%! +& |( @D

sinh x ∼ x

) (sinh x − x)

∼

x

√ √ O(sin( x) ln( + x)) = O(sin( x)) + O(ln( + x)) √ = O(sin x) × O( x) + O(ln( + x)) × O(x)

V 0=0=> @+A/ L N; + C &

∼ g

∼ f

) af ∼ af

)

f (g) ∼ f (g )

+ i: 2j = = a ∈ R )

f g ∼ f g

)

f /g ∼ f /g

G d T"& f '$ %+& @D θ

= O(f )

S

=f (ax)

2

=

2+

∼ aθ f (x)

( x)

lim

x→

− cos(x) = lim x→ x x

$

=

)

+ D!3

)

− cos x √ ) y = x + x

)

y = (sin x − tan x)

y = sin x − x + x ) y = sinh x

)

y = e(sin x−x) −

y = ln(cos x) ) y = +x −

) − cos x) ∼

×

) y = sin (x )

sin (x) (x) lim =

= lim x→ sin(x ) x→ x

mH T(

×

L 8$ %+& L ' P @D

#

x

× + × = + = √ √ O(arctan( x) ) = O(arctan( x)) √ = × O(arctan x) × O( x) =

& lim f (x)/g(x) @D!3 x→ P < g < f & S VD %Q 74 N 8 d& T ! "L VP L d < = ++a mH Tsin x ∼ x $

2 N

N +P

2 N

y=

)

y = ln( + x − x + x )

) y = arctan (sin (tan (arcsin x)))

'$ %+& < y = g(x) < y = f (x) S V+ /< y = f (x)/g(x) %./ | G d T"& D VP kN+/ =%! %+& < 4 [Q( T + 4 & x & D VP @;& & TW!H ^ & !H & -%! N$ =V+ !& M+/ / " n;

=

2 N

& TL . + & L VP L ! & G d T" ! d L S Tsin x ∼ x 8

+ i: ! y = g(x) < y = f (x) 4 =8$ %+& =f ∼ g V+ 24 ^ = lim f (x) = V+hS x→ g(x) =%! & D VP TL VP & LO " %! ^"<

< y

sin x − x ? x−x = lim = x→ x→ x x

= g(x)

>! 9

lim

%! +Q ^+.P & =%! && ^ V+ Q ^(! '$ %+& < L VP U.B L 1=1=> @+A/

=%! S

= f (x)

/& L s&<

VP \< +S8& L mH t $ TW&& s7l 4 | ! TV+!& %+& < 4 !H & L 8$ %+& L P @5Q; L VP L L T%! =" ! D VP

) (

) sin x ∼ x

) (sin x − x) ) 91

(I ) 6 4K "

∼ −

loga ( + x) ∼

x

x ln a

− cos x) ∼

)

tan x ∼ x

)

ln( + x) ∼ x

x

- . / 0*1 J

Tf (x) =

√

:*1; $

+x

f () = f ()

=

f ()

=

f ()

2d

+ i:

=

L < TL VP \< '. &

24 ^

+ x

ln( + x) ) lim x→ sin(x)

= ( + x)−/ = x= − − ( + x)−/ = x= ( + x)−/ = x=

x=

)

+x−

+x−

∼

−

x

x

+

x

∼

+x−

−

x

∼ −

)

x

ex −

−x−

x

− ···

xn n!

∼

)

y = ln( + x)

)

y = sinh x

)

esin

lim

x→

+x − +x − − esin x tanh x x

)

ln( + sin(x)) x→ esin(x) −

)

cosh x − x→ x

lim

lim

lim

x→

ln(cosh x) ln(cos x)

) lim

+ x x + x

(I ) 6 ( S/> 4/ ? )

+G& p L

(ex − ) ∼ x (ex − ex − e − x

y = cos x

)

y = tan x ) y = n + x

x

− x) ∼

−x−

x

−x−

x

∼

x

xn − ··· − n!

∼

xn+ (n + )!

L VP L S fke T%! & 7D/ L ' P TQ: ^

G d TV+ ! < lim

+, - ./(0

lim

− cos x − cos x

+ sin x − cos x + sin(px) − cos(px) x→ ln( + xex ) +x+x − ) lim ) lim sin x x→ ln(x + x→ +x ) ln(x + ex ) x→ ln(x + e x )

Q:

xn+ (n + )!

+ D!3

x→

)

lim

) lim

Q: TL U& L ' P ! +& n = 06=1=> @+A/ s! " +Q L VP ) y = sin x

x→

+x −

sin x − x x→ tan x + sin

2 N

lim

)

P ǱL & 24 ^ Tf (x) = e + i: ǱL & mH =f (n) () = ^&& < f (n)(x) = ex n ∈ N V n P x

x→

)

x

ln(cos x)

V+& V+ 09=1=> @+A/ '. & T^&&

lim

)

x→

I

ex −

−x

sin x

= lim

x→

x / = x

x " 4 B V ! Q< L S Q & < 7+ @+A/ TL VP Q: -+Q UD =w%! s7l ="& Q ' @+A/ 5 Td Z %Q R<

%3 W(& & TK+ _: L ! 2N @P & =" 5B ' j: L ^+.P U& + i: X B 2G I ( @ 43 @D!3 & =V . [5 K+ s+3 fkD/ y = f (x) limit(f(x),x=c) ! L x = c @;N U& ^ =V+ ! y = f(x) U& %! < $ @D!3 & < limit(f(x),x=c,left) 2 ! L {+ & Tx = c @;N =V+ ! limit(f(x),x=c,right)

i: (I ) 6 ( S/> 4/ < B x = y = f (x) U& L n + @D M +

24 ^ T%! !+H

9J

f (x) − f () − f ()x − · · ·

f (n) () n x n!

∼

f (n+) () (n + )!

:*1; $

- . / 0*1 J

!

24 ^ +l < true !H T& !+H 24 =. =& P false + & readlib(discont): discont(f(x),x) ! & =. ! f(x) U& G !+H N ^ : + B L n" & Z $ &

! L ∞ < & undefined ! L T"& " B< t $ =V+ =V+ ! infinity

+ B L n" & Z $ &

K+

limit(xˆ2 − 3 ∗ x + 1, x = 2) −→ −1

readlib(discont) : discont(1/(xˆ2 − 1), x = −2..2)

K+

K+ 1

limit((x + 1)/(2 ∗ x + 1), x = inﬁnity) −→

−→ − ,

K+

limit(exp(1/x), x = 0, left) −→ 0

readlib(iscont) : iscont(1/(xˆ2 − 1), x = −0..2)

K+

K+

limit(exp(1/x), x = 0, right) −→ inﬁnity

−→ f alse

cd

2

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http://webpages.iust.ac.ir/m_nadjafikhah/r1.html

=%! "
readlib(iscont): iscont(f(x),x=a..b)

N+N3

6)

2E/J YA

[a; b]

@L& &

f(x)

U& G !+H M+N3 &

BCD ?E /F GD H I:" @ %/J x = x =%+ : w" ' TP 0= =9 KL ,"

L & _+ < + < M @h j: ^ L vP ="& %54 < '_+: T2+ d P&

`b)-!

=%! ]KN x = x @;N y = f (x) V+ i: d L & B< f (x ) T^&& f (x) − f (x ) < |x − x | < δ ⇒ − f (x ) < ε x − x '.

J(K

∀ε∃δ∀x

S B< δ T (Q

P L & T^&&

G d T|x − x | < δ

< ε

= f (x)

+ i :

(f (x ) − ε)(x − x ) < f (x) − f (x ) < (f (x ) + ε)(x − x )

f (x) − f (x ) lim S =x ∈ Df T " & " B< x→x x − x ^ N < %! x y = f (x) V+hS df < f (x ) . & < + x = x y = f (x) M dx x V+P f (x ) := lim

x→x

8 <

<

f (x) = x −

f (x) − f (x ) x − x

x + i:

|f (x) − f (x )| < max {|f (x ) − ε|, |f (x ) + ε|} δ

" i: %! : T

ε δ = min δ , |f (x ) − ε| +

ε , |f (x ) + ε| +

\ ]K

5

& ^ =V+ Lld '. & M [5 & =%! +

< % ! U & y

|f (x) − f (x ) − f (x )(x − x )| < |x − x |ε

lim

x→

f (x) − f ( ) x−

= =

2

24 ^ Tx =

(x − x) − (− ) x− lim (x − ) = lim

x→ x→

&& d M < %! ]KN x = y = f (x) ^&& =f ( ) = T5 q%! 4 24 ^ Tx = < f (x) = |x| + i: f (x) − f () |x| = lim sgn(x) lim = lim x− x→ x→ x x→ x = y = f (x) mH w-$x B< ^

=%+ ]KN

2 N

S + |x − x | < δ i: L 24 ^ x = x @;N y = f (x) U& ^&& |f (x) − f (x )| < ε 2 =%! !+H ' & 3 + i: Y* 2Q H> ; 8 TO % Lld @;N S = % V+N s O L y = f (t) @74: & V+G& f (t) & t @p3Q }3 }3 p3Q %C! w'_+: [5 &&x G d T%!

2+ M:" " N,O H f (x) 2+ " I:" . :P(%Q x

y = f (x)

y

6

=

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KL #4 7G $ M*N

x

M*N

@L& GP s! %C! & %! && t = t @p3Q %! '$ : @L & < & x " L

& '_ : @L & P x ǱL & Ok< + _& {N V+"& " w =9 x

f (x) = ax + b

;x x − x & g(x) = f (x) − ax − b %D T E < 4 && T + x & x /< w<< ; & B< 5 T"& g(x) = x − x ^&& Tg(x ) "

L @L& D %C!

lim

x→x

x

f (x) − f (x ) = f (x ) x − x

@p3Q }3 p3Q %C! T^&& =%! L & %D % @;&

U& . + i: 2456 ; N L ! s =x ∈ Df < V V! f (x) s ^ qV+S p ]S (x , f (x )) < (x , f (x )) + x & x /< s ^ %Q =V+ lx x @;N y = f (x) U& . & c. s =

V w0=9 x < w =9 x < L T
y =

g(x) f (x) − ax − (f (x ) − ax ) = lim x→x x − x x − x f (x) − f (x ) = lim − a = f (x ) − a x→x x − x =

x

L @L& ^ 8 ++a x→x L L& ^ L ++a @

M & &&

+ w0=9 x L

lim

=

f (x ) = ax + b

x→x

=

w0=9 x

lim

x→x

lim x

v(x ) =

lim

x→x

@;N Q y = f (x) U& T5 =a = f (x ) T^&& T%! f (x) ≈ f (x ) + f (x )(x − x ) U& && kDN x = x '$ %+& ' << ; & %D 4 ;

=%!

2

& 3 % @;& " 24 & x L { & T % V+N + ' p3Q %C! =%! " +& f (x) = (x + )/(x + ) V+ D!3 ,< @p3Q d v( ) = lim

x+ x+

− = lim

=

(x + ) y = x + U& . & c. s @Q5 √ %! : Tp ^ & =V+& x = @p3Q @;N L s ^ V+ L TV+ D!3 s d {+" √ =]S ( , ) x→

x−

m =

lim √ x→

=

lim √ x→

=

lim √

x→

x→

2 N

M !P +D5 =9 8" : ^&& T%! 75 s ^ L (x , f (x )) @;N $ d& ^+$ [5 L =V+ D!3 s ^ {+" %! lx s {+" && x y = f (x) & l c. s {+"

+ = + x & x X" & T%!

x + − √ x− x + + x + + √ x− x + +

(x −

m =

x − )( x + + ) √ √

=

√

= lim = √ x→ x + + L %! 2DC p s @Q5 T^&& √ x − √ y = f (x ) + m(x − x ) = + √ =y = x + T5 x+

=

l

c. s {+" lim (lx s {+") = lim

x →x

lim

x →x

x →x

54 N <+H N

f (x ) − f (x ) = f (x ) x − x

y = f (x) U& . & c. s {+" & f (x ) T^&& =%! && x = x @;N < %! U& y = f (x) + i: (H -= ; ! X y = ax + b ; 5& & y = f (x) U& =x ∈ Df 60

@*N <# = 1

KL #4 7G $ M*N

V+ i:

⎧ ⎨

2 N

@;N Q f (x) = ln x U& V+P V+ B =V+ _& {N ; U& ' & x =

S S S y = f (x) < y = f (x) U& . < D!3 d M =+ V! . S! & D/ j: 2CfX L ! & = T
−x ( − x)( − x) ⎩ −( − x)

f ( −) = = f ( +) = = f (−) = = f (+) = = y = f (x)

x< ≤x≤ <x

f ( )

= =

ln x − ln x− ln x ln(z + ) = lim = lim z x→ x − z→ lim

x→

^&& wz = x − x ln x ≈ ln( ) + ( )(x − ) = x −

TM [5 L ! & P +&+& x = x y = f (x) U& M

f (x) − f ( ) x− x→− ( − x) − lim =− x− x→− f (x) − f ( ) lim x− x→+ ( − x)( − x) − lim =− x− x→+ f (x) − f () lim x− x→ − ( − x)( − x) − lim = x− x→ − f (x) − f () lim x− x→ + −( − x) − lim = x− x→ + lim

,

x = −

√ ) f (x) = x,

x = ,

)

f (x) = cot x,

x =

,

)

f (x) = arcsin x,

x =

,

)

f (x) =

)

f (x) =

x

,

x+ x−

π

x = , x = − ,

,

)

f (x) = x sin(x − ), x ) f (x) = , x+

) f (x) = cos x , sin x + x ) f (x) = , − x

. < . ! 2CfX ^ L =. V+! `0=9 8"

√

) f (x) = x ,

x = , x = , x =

π

,

x = .

%! < $ T M < $ M T n; d @ : V+.5 2j& . [5 M V+.5 C & L 2j& %! . w y = f (x) U& . w[Q 0=9 8" y = f (x) M U&

=(K 9!: 0

f (x) − f (x ) , x − x f (x) − f (x ) . f (x −) = lim x→x − x − x

f (x +) = lim

x→x +

M:" " 6 4 N,O H x y = f (x) " I:" :P(%Q x " >%Q 4 I:" " 4 / R :P(%Q 4 " :P(%Q 2 @ f (x) y = f (x)

3 & %! M D!3 W(& ^ L vP ' p ^ & ="& " & G & / =V+ _ nX DB V \<

#

6>

KL #4 7G $ M*N

)

(tan u) =

)

(cot u) = √ ) ( u) =

u cos u

@*N <# = 1

)

−u sin u u √ u

)

(arccotu) =

)

√ ( n u) =

−u +u

u

(sinh u) = u cosh u

)

(tanh u) =

)

(cosh u) = u sinh u

)

(arcsin u) =

)

(arccos u) =

)

(arctan u) =

)

ln

)

u

=%! C & ^. C & +A/ ^ 2DE

−u

y

−u

=

9" <= . :P(%Q y = g(x) z = f (y) 4 x = x Y* (%U-I % 4+

= g(x)

(f ◦ g) (x ) = g (x )f (g(x ))

V [5 L ! &

u −u

f (g(x)) − f (g(x )) x − x f (g(x)) − f (g(x )) g(x) − g(x ) × lim = lim x→x x→x g(x) − g(x ) x − x f (y) − f (y ) = lim × g (x ) = f (y ).g (x ) y→y y − y x→x

`b)-!

y = g(x ) &

B &

9" .y = f (x) 4 :P(%Q x = x 4 / :PJT/M* 4 " :P(%Q y = y x = f (y) <= P/Q FG Y* % 4+

y = f (x)

ex − ey ey−x − = ex lim y→x x − y y→x y−x z e − = ex × = ex ex lim z z→

−

f − (y ) =

f f − (y )

L +S M & Tf − (f (x)) = x 8 & B & A7S" & < Tx = x = f −(y ) @;N x { & ;& ^ ^+:X V 0=0=9 @+A/ L !

: Tw x 2DE & =z = y − x %! " i:

,: 2DE =au = eln(a ) = eu ln a " B %! V >=0=9 L ! & Tw Ix u

f − (f (x )).f (x ) =

u −u u = = (arccos u) = cos (arccos u) − sin(arccos u) −u

2

^. C & ! 2DE =sin x = − cos x L 2 =VK! &

2

=%! " 2DE V8

2

lim

T =0=9 L w>x %./ & B &

A7S"

(f ◦ g) (x ) = lim

d ! 2DE < . 2DE Q: ^ L & : w0=0=9 &&x w0x 2DE & =VK! & =(ex ) = ex " %&E %! =

(f ± g) (x ) = f (x ) ± g (x )

(f g) (x ) = f (x )g(x ) + g (x )f (x ) f f (x )g(x ) − g (x )f (x ) ) (x ) = g g (x )

cosh u

u ) ln u + u ± = u ± ) (uv ) = v ln u + v u uv u

(ex )

)

−u

u +u + u = − u

y = g(x) 4 y = f (x) 9" A<= .a ∈ R 4 :P(%Q

) (af ) (x ) = af (x )

u √ n n un−

)

x = x

(cos u) = −u sin u

=V+

⇒ f − (y ).f (x ) =

f − (y ) {

& Q5 ^ %! :

D Td P Q<B .P & +A/ ! ^ ="& M U! &

*"/+ v(x) 4 u(x) U6

V

) (au ) = u au ln a

(x cos x) = (x) cos x + x(cos x) = cos x − x sin x

69

)

(ua ) = au ua− ,

)

(ln u) =

u u

Z * 4K

A7/V I:" . :P(%Q )

(eu ) = u eu

)

(loga u) =

)

(sin u) = u cos u

u u ln a

@*N <# = 1

KL #4 7G $ M*N

V T =0=9 L w9x %./ & B &

y = arctan x + arctan x sin α sin x ) y = arcsin − cos α cos x x − x ) y = − x ( + x)

)

)

y = x(x − ) · · · (xn+ − n)

+ D!3 4 f (x) f (x) =

x x x x

x− x − x − x −

/x x /x x

2%

⎛ ⎝

=

+&+& y P

)

y = φ(x)ψ(x)

y = f (ex ) + ef (x)

f (x) =

2 U& +P 2

x |cos (π/x)|

x = x=

(x)(x − ex ) − ( − ex )( + x ) (x − ex )

=

S S

+

2 N

x+ x +

x+ x + (x + ) + x +

+ x+ x + ×

x+ x + x

+ x+ x + ×

%! XN x = @;N L (Q G.P P =%! ]KN x = y = f (x) Q< T M d + D!3 f (x−) < f (x+) TL L ' P

24 ^ Ty =

+ x +

a x b a x b b

x

a

V+ i:

2 N

ln y = x(ln a − ln b) + a(ln b − ln x) + b(ln x − ln a)

f (x) = |x| sin(πx) ) f (x) = − e−x +x ) f (x) = arctan ) f (x) = sin x −x x + x ) f (x) = ) f (x) = | ln |x|| x

)

+ = y

= ln a − ln b −

y

a b + x x

V +GN L mH <

=y = y

b−a a + ln b x

%D y = f (x) L L L ' P

2+ U6 . F (x, y) = c W 7/V y = f (x) :P(%Q y 4 x ? " X% 4 *+ z = F (x, y) 7/V I:" @ c 4 " 25O FG/ Y* % 4+ "

y = −

=

+ x+ x + (x + x + ) x+ x + + x+ x +

y = logφ(x) ψ(x) ) y = f (sin x) + f (cos x) x x ∈ Q S f (x) = U& +P x∈ / Q S =%! ]KN x = ǱL

( + x ) (x − ex ) − (x − ex ) ( + x ) (x − ex )

⎞ + x+ x + ⎠ =

=

) &

=

V T =0=9 L w0)x %./ & B &

)

+x x − ex

2 N

√ √ ) y =x+ x+ x

Fx Fy

? z = F (x, y) 2+ >%Q ," " 7 F G= 2+ >%Q ," " 7 F 4 y G/ S U6 4 x 2 @x G/ S U6 4 y ? z = F (x, y) x

+x−x −x+x

)

y=

)

y = sinn x cos(nx)

)

y = sin(sin(sin x)))

y

)

y=

x+

√ x+ x

66

+G& M x &

y = tan(/x) ) y = a + x

)

x y= a −x ) y = + x −x

)

)

x

ex

y = ex + ee + ee

) y = (x + )(x + ) (x + ) + ln + ln ) y = ln x x x

2DE < 7B h_B 2N 3D +A/ ^ =. V+P

!

KL #4 7G $ M*N

=

O4 5# <LM*N

dy ψ (t ) dt t=t = = dx φ (t ) dt t=t

y−y t−t lim x−x t→t t−t

M Tx

=%! . P& <

2

Ty = sin t < x = sin(t) S

y =

( sin t) = (sin(t))

G d Ty =

y

2

cos t cos(t)

−

√

t<x=

−

√

y = −

t S

2 N

y

√ √ ( − √ t t) ( − t)−/ = √ = ( − √t) ( − √t)− / − t (− t− / )( − √ t)−/ = √ −/ )( − t)− / (− t √ / / t − t √ = t − t

y

) ) )

x = arcsin

)

t

+t

y = at( − cos t) , y = arccos

+t

=

++−− = − ++ −+ = f (x)

M %! 7;

xy + y x = xy x

& %D

= f () (x) = (f (x))

y (n)

= f (n) (x) = (f (n−) (x)) (y = f (x)

)

x + y = tan(x + y)

)

ln(x + y) = x + y

y = f (x)

M %! 7; 8 d & < √ √ M = ( π, π)

x sin y − y cos x = π,

)

(x + y)x−y + (x − y)(x+y) = , M = (, )

( ) J FG/ Y* #

dy dy ψ (t ) dt t=t = = dx dx x=x φ (t ) dt t=t

= f (x) = (f (x))

y ()

8 d & < Tx

U6

y = ψ(t) 4 x = φ(t) 7/V x ," *+ .x = φ(t) . :P(%Q t = t ψ 4 φ 9" @ " x ? y = f (x) 7/V I:" .y = ψ(t) 4Y* 4 " :P(%Q x = x

]KN y = f (x) U& + i: y = f (x) . & d M 24 ^ T"& [5 & 24 & TV+P y

( , )

)

y = f (x)

L3 4! 9MJ(K

2 N

=

TM @;N

y = b sin t + t

x = a(t − sin t),

+ xy + yx = x − y

) y sin x + x sin y =

y = cos t

x = a cos t,

Fx cos(x + y) − yxy− =− Fy cos(x + y) − xy ln x

x + y + xy − x + Fx − =− Fy (,) + xy + x − + y

& %D y T"& %De C b < a 4 + D!3 x & %D y M

= xy

+ i: =+ D!3 ( , ) @;N x & %D + < x + xy + yx = x − y =

M Tx

−

) x = sin t,

2 N

+ i: =+ D!3 + < F (x, y) = sin(x + y) − xy =

cos t cos(t)

x + − y Fx y − x =− = Fy + y − xy y − xy

x & %D y M Tsin(x + y)

24 ^

+ i: =+ D!3 x & %D y + < F (x, y) = x + y − xy =

y = −

=

2

+ y = xy

^&& T%! !+H T]KN U& P $ A7S" ^&& < limt→t x(t) = x(t ) = x

== =

n M )

dy dx x=x

6F

=

lim

x→x

f (x) − f (x ) y(t) − y(t ) = lim t→t x − x x(t) − x(t )

O4 5# <LM*N

KL #4 7G $ M*N

P ^+$ L %! : T^&& =S _

V T(ex ) = ex 8 & B &

A x+a

24 ^ Tga(x) := x + a S =V+G& M V+ & ga (x)

=

ga (x)

=

ga(n+) (x)

=

(− )n (x + a)−(n+)

=

(− )n+ (x + a)−(n+

V T(ax ) = ax ln a 8 & B &

f

G d Ty = sin x 4 π , = cos x = sin x +

y

)

y

=

g (x) −

g (x) +

π = − sin x = sin(x + π) = sin x +

(x + )

− (x + )−(n+) +

(x + )−(n+)

y (n)

= =

2# N

dy (n−) = = dx

^&& Ty = − x y

x y

− y − x

− y − x

<

− y)

y

=

(n)

x

=

− , x

=

y (n−) =

=

(− )n− (n − )!( − n)

xn

=

(− )n− (n − )!(n − )

=

(− )n− (n − )! xn

xn

(− )n− (n − )! xn−

f (x) =

2+ N

x(x + )(x + )(x + )(x + )

XX

= x(x + )(x + )(x + )(x + ) B A C D E = + + + + x x+ x+ x+ x+

T d %!& 2DC . ! < ^ :S } |( L mH + < / d %!& 2DC < T T T C < D = − TC = TB = − TA = V+S / O& < N ^ %! : =E = : & @! H . & y = f (x) {+ ^ & < T

cos t − t sin t dy = dx sin t + t cos t

^&& dy = dx

=

2 N

=+ D!3 _ O& < %! %.! & p ^ & = V+ i: T^&& =w" B &x V+

2% N

=y V+ B p ^ & =

y

, x

U& n @D M

N y & %! :

%&7; Ty = t cos t < x = t sin t 4

y =

y

dy (n−) dt dx dt

x − y − y − x (x − y )(y − x) − (yy − )(x − (y − x)

=V+P / −

=

== =

=y %&7; Tx + y = xy + + i: V+ B p ^ & =

y

y

−(n+)

G d Ty = ln x 4

−(n+)

+ (x + )

d dy = dx dx

=

(n) g (x)

− g (x) + = (− )n n! x−(n+) −

π y (n−) = sin x + (n − ) π π = sin x + n = cos x + (n − )

y (n)

g (x)

2 N

== =

(x)

2 N

(ax )(n) = ax (ln a)n

+ (n)

(ex )(n) = ex

(x + a)− = −(x + a)− ,

−(x + a)− = (x + a)− ,

== =

2

dy dt dx dt

6I

KL #4 7G $ M*N

< u = e x i: & Ty

y

( )

=e

x

cos x

O4 5# <LM*N

2 N

4 V Tv = cos x

) = (e x )(k) (cos x)(−k) k k= = (e x )() (cos x)() + (e x )() (cos x)() + (e x )( ) (cos x)() + (e x )() (cos x)( ) + (e x )() (cos x)() + (e x )() (cos x)()

=

sin t + t cos t

4

v(x) u(x)

(n)

(vu)

) y=

√ x, n =

x , n= −x

)

y=

)

y = sin x ln x, n =

ex , n= x

)

y=

)

y = x sin x, n =

)

cos(x) y= √ , n= − x

)

y

y=

ax + b cx + d

)

y= √

)

y = sin(ax) cos(ax)

) y = (x + x)e−x

)

y=

= f (x)

)

n ) n (k) (n−k) (vu) = u v k k= n ) n (k+) (n−k) u = v + u(k) v (n−k+) k k= n n ) n (k+) ((n+)−(k+)) ) n (k) ((n+)−k) = v + u u v k k k=

=

= uv (n+) +

=

n=

)

sin(x + y) + cos(x − y) − xy,

n=

)

x = a(t − sin t), y = a( − cos t), n =

)

x = sin t, y = cos t,

n=

)

x = arcsin t, y = arccos(t ),

n=

y ()

2,

Q< T B< f (n)() + %&E %+ B x

n

sin ( /x)

x = x=

n+ ) k=

n+ k

n k−

k=

n u(k) v ((n+)−k) + k

u(k) v (n+−k)

i: & Ty = x ln x 4

n=

y = xy + y x − x + y ,

f (x) =

n+ ) k=

U& n @D M TL L ' P

n ) n n (k) ((n+)−k) (k) ((n+)−k) + u v u v k− k

2

)

f (n+) ()

k=

n+ ) k=

x(x − )

xy + x y + x − y,

u(k) v (n−k)

(n+)

− x

y = x sin x

k

fkD/ n = ǱL & V8 =V+ ! n & ǱN ! L A7S"

G d TV+ i: %! n & V8 S =%! " 2DE

+ D!3 n @D L M TL L ' P )

=

n ) n k=

+(

n @D M TL L ' P
−( cos t − t sin t)(cos t − t sin t) (sin t + t cos t)

H5;[ ) /&

= ( )(e x )(− sin x) + ()(e x )(cos x)

=

×

cos t − t sin t sin t + t cos t (− sin t − t cos t)(sin t + t cos t)

2"/+ +/V <= . % " N" n + >%Q

)(e x )(sin x) + ( )(e x )(− cos x) +()( e x )(− sin x) + ( )(e x )(cos x) − e x sin x − e x cos x

d sin t + t cos t dt

=

S S

V Tv = ln x < u = x

) )() (ln x)(−k) (x k k= )() (ln x)() + )() (ln x)() = (x (x () () ( ) + (x ) (ln x) + (x ) (ln x)() )() (ln x)() + (x − = ( )(x ) + ()(x ) x x − +()(x) + ()() x x

=

+( )()(ln x) =

61

2

=%! . P& <

x

?#*PG QP

KL #4 7G $ M*N

'8QK" ' 4 x

< "D B f (x )

# f (x) =

)

"!6

− <x<

⇒

⎪ ⎩ <x − <x

⇒

⇒

<

⎪ ⎩

<x

−

(x ln x)

= n! ln x +

n ) k=

(n) x +

(− )n /x e , xn+

=

(x = ) ,

k

(x > )

(− )n n! (x + )(n+)/

< %! U& ' y = f (x) + i: y = f (x) U& V+hS 4 =x ∈ Df " %: $ δ %De C %! x = x f (x) ≤ V+"& " T(x − δ; x + δ) x P ǱL &

y = f (x) U& V+hS 4 T& 24 & =f (x ) $ δ %De C %! x = x V+"& " T(x − δ; x + δ) x P ǱL & " %: ._ z 5 & @.7 =f (x ) ≤ f (x) =V& 8& y.++

F0S

(x − )

)<"

)<" F0C

− − = x (x − ) > > = − =f

=

V7C ,G < + P& ^ . L 8 =%! L! +& @Q@ < @Q@ TL<
T;N < ^+& P ,j P+ . ^+& L Tfke = ^ ="& / d ,X S + ,D &

+a ' U& '. & Q@ ^ L ! W(& =V+ 5Q; .

^&& f (x)

+ %&E L L ' P

<!(NF ON

ǱL & L T%! y = f (x) U& D .++ ' x = w V x ∈ − ; + P <x< ⇒

(n)

S S

2

⇒ x (x − ) ≤ = f ()

⎧ ⎪ ⎨ x−

(n)

n

x = x=

Tf (x) = f (x) & : < LO " +P =f (x) = Aex A %&E C L & %! d

x− < x <

⎧ ⎪ ⎨ x<

xn− e/x

%!

× sin{(n + )arccot x}

2

)

)

U& 5 & f (x) = " T%Q ^ =+G& p @+A/ & B & mH =%! x(x− ) = < x − x = 3& N y = f (x) U& Q. D TO& 24 ^ = P x = < x = T%! y = f (x) U& D ._ ' x = w[Q V x ∈ (− ; ) P ǱL & L f (x) = x − x

e−/x

@;N V+hS y = f (x) U& ="& 4 & && B< 24

∈ R

2_

]KN P(Q @D P L L U& +P

< %! LO " ' f (x ) = " " B " ^ ;N %! ^8. 5 ^ & =%+ :

! ="D D @;N ' Q< r4 y = f (x) M S %! ^ V8 ^ L

G d T"& 4 && < " B< < "D B x = x

F < !

=

2 N

" T%Q ^ =+G& p f (x) = x U& U& 3& @;N ^&& T%! x = 5 & f (x) = @;N ' %! ^8. T"& x = p

y =

S

x = x

6J

./$' &/K

(G .I[ \W

< "& D x = x f (x) =f (x ) = G d T"& ]KN

?#*PG QP

KL #4 7G $ M*N

T"& |
G d ._ @;N ' x = x G d Tf (n)(x ) < S w[Q =%! y = f (x) U&

U& @;N ' B< ?+P & ;N ^ ="& _+ d L T%+ x < ⇒ x < ⇒ f (x) < f () x > ⇒ x > ⇒ f (x) > f ()

U& .++ @;N ' x = x G d Tf (n)(x ) > S w =%! y = f (x) f (x)

=%+ : " ' f (x) = " %&E ^

2

S ./$' &/K (G 2> ' \W < "& ]KN x = x @;N L G.P ' y = f (x) G.P y = f (x) %fC @N & G d Tf (x ) = V x

+ x − x + i: f (x) = " L T^&& =f (x) = − x 24 ^ U& ^ 3& @;N TmH =x = S + %Q ^ T =%! x = /

y =

=

x<

⇒ f (x) =

− x >

−

=

x>

⇒ f (x) =

− x <

−

=

U& L N & ._ @;N ' x

8 + @;N ' x %+ %! ._ %! .++ %+

T^&& =%! f (x) < %! : C n ∈ R + i: +x+

x

+ ···+

&/K

x = x

xn− − ex (n − )!

+x+

x

+ ··· +

(G

2> '

\W

!

@;N L G.P ' y = f (x) S ./$' < T"& n @D M f (x ) = f (x ) = · · · = f (n−) (x ) =

+l @;N ' x T"& : n S G d Tf (n) (x ) = < f (n) (x) %fC @N & T"& |
24 ^ fn (x) =

%Q [Q |

=

2 N

fn (x) =

f (x) %fC f (x) %fC x > x ǱL & x < x ǱL & + + − + + − − −

xn− − ex = fn− (x) (n − )!

%fC %fC 8 + f (n) (x) f (n) (x) %Q @;N ' x x > x ǱL & x < x ǱL & %+ + + [Q fn (x) = fn (x) = · · · = fn(n−) (x) = , fn(n) (x) = −ex %! ._ − + %! .++ + − | (n) x = G.P y = fn (x) %fC " p f %+ − − x = @;N TF=9=9 @+A/ M&; T^&& =%! wx %&E ./$' &/K (G 2> ' \W & " =%+ y = f (x) U& T"& x = x < @D M y = f (x) U& S 24 ^ Tf (x ) = < f (x ) = ._ @;N ' x = x G d Tf (x ) < S w[Q 24 ^ =f (x) = xe−x + i: =%! y = f (x) U& TmH =x = ]Q < e−x − xe−x = + Tf (x) = x = L %! 2DC y = f (x) U& Q. U& .++ @;N ' x = x G d Tf (x ) > S w 24 ^ T =%! y = f (x) %Q ^ T =%! d @ ' x =

2 N

f ( ) = {−e−x + xe−x }x= = −e− <

U& L

e

−

N & ._ @;N '

x =

&/K

(G

2> '

n @D M

\W

&

U& S ./$' f (x ) = f (x ) = · · · = f (x ) = T"& x = x T"& : C n S 24 ^ Tf (n)(x ) =
T^&&

=%! y = f (x)

F)

y = f (x) (n−)

?#*PG QP

KL #4 7G $ M*N

y = f (x) .++ f (x )

< %! I & y = f (x) .++ @;N &N x S ^ = I & = M7; wD

fn (x) =

2

fn (x)

e−x

xn− xn − n! (n − )!

e−x

U& ._ ' x = G d Tn = S T^&& f () = −e− < < f () = f () = L T%! y = f (x) = U& +l @;N ' x = G d Tn = S =%! y = f (x) U& ._ @;N ' x = G d Tn = S f () = f () = f () = f () () = L T%! y = f (x) n N ! & {+ ^+.P & < f () () = −e− < < =. 3&

XX

U& @7 & @+A/ & B V+ .C L \< & I = [a; b] ⊆ Df @L& & =%! !+H [a; b] & y = f (x) d V+ M+N3 w[Q =V+P T& ^+$ S XN < !& (a; b) & y = f (x) ]KN w 4 && < B M < B< M d =V+ ( w 3& N x =%! d %!& 3& N y = f (x) U& N w| =V+ D!3 wx @7 N & d T D!3 f (b) < f (a) N w ._ ^ S_& T N w|x @7 d %!& =V+ fC I & y = f (x) U& .++ ^ 8$ <

L U& L ' P ) f (x) = (x − )

2

+&+&

f (x) = (x − ) √ ) f (x) = x ln x

f (x) = (x + ) e−x

)

f (x) =

)

f (x) = |x|e−|x−|

)

f (x) = x(x − ) (x − )

)

f (x) = cos x +

x − x + x + x +

#

)

)

)

< f (x) = x + i: !+H [− ; ] & y = f (x) U& %! ^"< =I = [− ; ]

" @Q@ mH ="& ]KN (− ; ) & < < L d wQ. x 3& N < %! B 4 S ?+P f (x) = x ln T =wf (x) = T5x T = I & 3& @;N ?+P y = f (x) T+ =".

xn n!

xn− xn xn− − − e−x = (n − )! (n − )! n! xn− xn xn− xn− () − − − e−x fn (x) = n! (n − )! (n − )! (n − )!

Y:< ./$' $ 73 ) /E

=%!

+ ··· +

fn (x)

< %! U& y = f (x) + i: ^ =%! !+H [a; b] & y = f (x) < I = [a; b] ⊆ Df ="& B I & y = f (x) .++ < ._ T24 ' 8 < P b < a N I & y = f (x) P ="& (a; b) y = f (x) 3& @;N

& y = f (x) ._ T^&& =f () = < f (−

=

2 N

I

x

2 N

n

24 ^ I = (; ] < f (x) = + i: x I & y = f (x) < %! I & y = f (x) .++ @;N ' x = M7; ._ I & y = f (x) ="& f (x ) = &&

y = f (x)

+x+

=%! − x e−x = 5 & f (x) = " 24 ^ n! ="& p U& 3& @; x = +

_+ < f (x) = sin x + i: π y = ._ @;N ' x = 24 ^ I = [; π] π = & && I & y = f (x) ._ =%! I & f (x)
&

+ i:

)

f (x) = ex sin x

cos(x)

f (x) = xm ( − x)n ,

; n, m ∈ N

U& y = f (x) + i: Y:< ./$' P ǱL & " %: $ x ∈ I S =I ⊆ Df < %! @;N ' x " S G d Tf (x) ≤ f (x ) x ∈ I y = f (x) ._ f (x ) < %! I & y = f (x) ._ & " %: $ x ∈ I S T& 2j& =V+ I & ' x " S G d Tf (x ) ≤ f (x) x ∈ I P ǱL

)=

&& d .++ < && F

KL #4 7G $ M*N

RG $ S8#O T"$ 7 <CD '

PF # Q7!L R # 6 9@A

2 N

=I = [− ; ] < f (x) = |x − x + | + i: @ !+H U& < {+ L T%! !+H y = f (x) U& @Q@ TmH ="& x → |x| < x → x − x + x P L& T =%! B I & y = f (x) < f (x) = (x− )sgn(x − x+ ) V Tx − x+ = wx = < x = T5x x − x+ = x P L& x − = + Tf (x) =
%

@ & @L& & y = f (x) U& S ) ^ =f (a) = f (b) < "& ]KN (a; b) @L& & T !+H =f (c) = B< c ∈ (a; b) ' / 24

[a; b]

,< @+A/ !P +D5 w[Q >=9 8" R SO @+A/ !P +D5 w

`b)-!

V+S p %Q ! =f (x ) > f (a) B< x ∈ (a; b) ' /O w[Q =f (x ) < f (a) B< x ∈ (a; b) ' /O w =f (x ) = f (a) x ∈ (a; b) P ǱL & w|

U& T L ' P ! +&+& I @L& & y = f (x) ) f (x) = x − x + ,

T%! !+H [a; b] & y = f (x) $ T ! P y = f (x) @Q@ w =I=>@+A/ & M&;x T^&& y = f (x) ._ T,< %Q =%! B [a; b] & TmH =P (a; b) L& @L& < %! S_& f (a) L [a; b] & & B< c ∈ (a; b) ' w >=9=9 @+A/ M&;x ="& w[Qx %Q +D" wx %Q =f (c) = d ǱL N . d M ^&& < %! %&E U& w|x %Q 2 =%! 4 (a; b)

)

f (x) = x +

)

f (x) =

)

f (x) = x (x − ) ,

x

,

I = [−; I=

]

− x,

;

I = [− ; ] I = [− ; ]

@D!3 \< +S8& & L < L ' P + %&E T{! L& & {! 5& M7;

G d T ≤ x ≤ < p−

S T,< @+A/ M&; 2456 ; Tf (a) = f (b) S _+ < "& ]KN (a; b) & < !+H [a; b] & 3 & c. s B< c ∈ (a; b) ;N G d & =%! P x 3 L (c, f (c)) @;N y = f (x) U& =" B [Q`>=9 8" y = f (x)

S w6

≤ xp + ( − x)p ≤

G d T ≤mx n≤ < < m, n S wF

x (a − x) ≤ m

n

m n am+n (m + n)m+n

=|a sin x + b cos x| ≤ a + b b P < a P ǱL & wI x + = ≤ ≤ x P ǱL & w1 x +x+

2

& f (x) U& & ,< @+A/ =+ M+N3 [; ] & x − x @;& & y = f (x) mH T%! 7.B $ y = f (x) $ = %! ^"<
._ ad

= bc

& f (x)

=

ax + b cx + d

+P wJ =.++ <

+ ^++5 X x + px + q q < p { w ) ="& " x = @;N 6 && .++ > S @;N f (x) = −x/x xx ≤ S +P w < f (x) %fC Q T .++ x = =%! 8 x = vX F0

RG $ S8#O T"$ 7 <CD '

KL #4 7G $ M*N

2 N

' f (x) = x + x − U& + %&E

2 N

< a = − ǱL & f (x) = − x U& V T− ≤ x ≤ P ǱL & T^ B< & Q< " 4 ,< @+A/ V8 & {7; ^ Pg / =f (x) = -%+$ Q< T%! !+H [− ; ] & y = f (x) %! ^"< = @+A/ s" mH = B< x = f (x) = − x−/ =%! A T5 = + /& ,<

f (x) = (x − )(x − x + )

= P N+N G.P < < ,< @D 2N 5& S +P w ) W @;N < P ^+& G d T"& !+H =%! [;C @;N ' /O @ & @L& & y = f (x) U& S ]-@[ 24 ^ T"& ]KN (a; b) L& @L& & < !+H [a; b] B< c ∈ (a; b) ' /

= < %! wR &x !+H y = f (x) TOk< =

f ()f ( ) = (− )() = − <

f (b) − f (b) = f (c)(b − a)

Tx ∈ [a; b] ǱL & TV+ i:

b =

U& M P + %&E wJ

^ = ' /O [; ] @L& y = f (x) ]Q Tf (a) = f (b) = < a, b ∈ R S $ T"& & j3 =f (c) = B< c ∈ (a, b) T,< @+A/ && G d =%+ ^8. c + = T5

`b)-!

g(x) = (b − a)f (x) − x(f (b) − f (a))

]KN (a; b) & < !+H [a; b] & y = g(x) 24 ^ @+A/ && T]Q g(a) = g(b) = bf (a) − af (b)

= B ' (;

g (x) = (a − b)f (x) − (f (b) − f (a))

d L & B< c ∈ (a; b) T^&&

) @L&

f (x) = x + x − x −

& ,< @+A/ ! w0 =+ M+N3 [− ; ] @L& <

@L& <

& ,< @+A/ ! w> π π =+ M+N3 ;

f (x) = ln(sin x)

f (b) − f (a) = f (c)(b − a)

@+A/ s" [− ; ] @L& < f (x) = |x| U& d w9 -$ - P ,<

=%! . P& <

2

xex = +P w

f (x) = + xm (x − )n U& M @D!3 <& w6

@L& /O U& ^ M + %&E Tn, m ∈ N d = ' (; )

S TR SO @+A/ && 2456 ; ! c ∈ ' G d T"& ]KN (a; b) & < !+H [a; b] & f (x) y = f (x) . & c. s {+" B< (a; b) (b, f (b)) < (a, f (a)) N ^+& 4< s & && x = c @;N =" B `>=9 8" & ="& y =

;N P P M y = f (x) U& + i: wF =x→a+ lim f (x) = lim f (x) ^+t.P < %! (a; b) @L& L x→b− B< c ∈ (a; b) 24 ^ + %&E =f (c) =

2

" < f (x) = x + x & R SO @+A/ =V+ M+N3 [− ; ] @L& !+H [− ; ] & mH T%! 7.B $ y = f (x) $ /& R SO @+A/ s" T5 =%! ]KN (− ; ) & < "& " B< c ∈ (− ; ) & T+ ="& T =f (c) = T5 f () − f (− ) = f (c)( − (− ))

@& c = =c = ± ^&& Tf (c) = c + = / (− ; )

"!e( &!'P0U CE ! n P . + %&E wI "& (−

; ) @L& j3 < N+N d Pn (x) = n {(x − )n } n! dxn

&!'P0U CE ! n

P . + %&E w1∗ "& N+N n x

Hn (x) = (− ) e

F>

dn −x e dxn

3 )E

KL #4 7G $ M*N

RG $ S8#O T"$ 7 <CD '

2 N

=| arctan a − arctan b| ≤ |a − b| b < a P L & w6 =

b−a < ln b

b b−a < a a

G d < b < a S wF

G d < β ≤ α < α−β α−β ≤ tan α − tan β ≤ cos β cos α f (x) =

√ x

c. V+& y = x 3 & ;N B = (, ) < A = (− , − ) N ^+& 4< < & ;N d ="& L A @;N ,X " B %! : Tp ^ & && mH =%! x = & B @;N ,X < x = − &

@L& < f (x) = x U& & R SO @+A/ %! : TF=6=9 B< c ∈ (− ; ) T5 =V+ ! [− ; ] Tf (c) = + =f () − f (− ) = f (c)( − (− )) $ < x = ± ^&& =f (c) = c V+ Q 2DC p @;N mH T / (− ; ) @L& c = =( , f ( )) = ( , ) L %!

π

S wI

U& & R SO @+A/ $ +P Y+ w1 =%+ Y+34 [− ; ] @L& <

S U& d wJ S - r4 R SO @+A/ s"

[; ] @L& < f (x) =

≤x<

x

/x

≤x

x, y ∈ R P L& V+ %&E

& y = g(x) < y = f (x) U& S 2W/' ǱL & T"& ]KN (a; b) L& @L& & T !+H [a; b] @ & @L& Tg(a) = g(b) < (f (x)) + (g (x)) = x ∈ (a; b) P B< c ∈ (a; b) ' 24 ^ f (b) − f (a) f (c) = g (c) g(b) − g(a)

V+ i: %! :

| sin x − sin y| ≤ |x − y|

C < a < b < f (x) = sin x V+ i: Tp ^ & [a; b] @L& < y = f (x) U& & R SO @+A/ L ="& (Q T%! N f (x) = sin x U& $ =V+ ! ]KN R & f (x) = cos x $ < %! !+H R & + /& p @ < U& & R SO @+A/ s" T]Q =%! B< c ∈ (a; b) T^&& =%!

`b)-!

h(x) = (g(b) − g(a))f (x) − (f (b) − f (a))g(x)

sin b − sin a = (cos c)(b − a)

(a; b) L& @L& & < !+H [a; b] @ & @L& & y = h(x) U&

T:X L =%:S M7; / ;& ^+:X L , : | sin b − sin a| ≤ |b − a| + < | cos c| ≤ =b = y < a = x " i: %!

=h(a) = h(b) = g(b)f (a) − g(a)f (b)
2

=%! . P& <

U& & " @+A/ # =+ M+N3 [− ; ] @ & @L& < g(x) = x − < [− ; ] & y = g(x) < y = f (x) U& %! ^"< = < g (x) = Tf (x) = x T ]KN (− ; ) & !+H "& c ∈ (− ; ) & mH =g() − g(− ) = = f( =c = = c 5 f (c) = ) − f (− )

g (c) g() − g(− ) f (x) = x

π

≤ x ≤

M+N3

[ ; e]

@L& <

[; ]

@L& < f (x)

f (x) = ln x

= arcsin x

& R SO @+A/ w =+

U& & R SO @+A/ w0 =+ M+N3 U& & R SO @+A/ w>

f (x) =

( − x )/ /x

≤x≤ <x

S S

=+ M+N3 [; ] @L& < L ! & < L ' P T1 9 2. + %&E R SO @+A/

2 N

P ǱL & +P √ = x ≤ sin x π Tf (x) = x V+ i: T{Q; ^ 2DE & = y = g(x) Ty = f (x) " @+A/ L < g(x) = sin x

V

2 N

py

F9

p−

G d T < p < < y < x S w9

(x − y) ≤ x − y p ≤ pxp− (x − y) p

345 U1#5 7 M*N . / 0*1 ,

KL #4 7G $ M*N

C & =R = (−∞; +∞) < [a; +∞) T(a; +∞) T(−∞; b] %! 54 [a; b] & y = f (x) /< < /< ,e ="& 54 (a; b) N . y = f (x)

c

d & f (x) = x − x U& P74: ! =V+ ( %! Q<_ < 54 f (x) %fC %! : T%! ]KN y = f (x) $ & Tf (x) = − x T =V+ !& −∞;

@L& & y = f (x) T^&& =%! ; +∞ & < %De

" B =%! Q<_ ; +∞ @L& & < 54 −∞; =54 < %! Q<_ x = @;N y = f (x)

√

≤

@L& < g(x) =

x

Tf (x) = x U& & " @+A/ w =+ M+N3 [ ; ] @ &

=arctan x ≤ arcsin x G d T ≤ x ≤ S + %&E w0 < f (x) = x U& & " @+A/ $ +P Y+ w> -%+ B &/ [− ; ] @ & @L& < g(x) = x

234 S 0!4 6 J(K - ./(0

*

=%! U& ' y + f (x) + i: ! %! 54 x = x @;N y = f (x) U& V+hS 4 x < x < x + P ǱL " %: $ >

" x − < x < x P ǱL & < f (x ) < f (x) V+"& " U& V+hS 4 T& 2j& =f (x) < f (x ) V+"& %: $ > %! Q<_ x = x @;N y = f (x) f (x ) > f (x) V+"& " x < x < x + P ǱL & " =f (x) > f (x ) V+"& " x − < x < x P ǱL & <

< 5N @;N w T3 @;N w[Q 9=9 8" ; @;N w| =V+&+& VP L ! ^ +( & "< & y = f (x) U& V+hS 4 ! ! 54 x = x @;N d M %! x = x @;N V+hS x = x @;N d 4 < T"& 4 T,5 +& & ="& Q<_ x = x d M 3 . & c. V+hS 5N x = x y = f (x) < " U/< 3 L x = x @;N y = f (x) U& s V+hS 3 x = x @;N y = f (x) 4 3 O& x = x @;N y = f (x) U& . & c. x = x @;N y = f (x) V+hS 4 =S U/< && ! s & ;N d L G.P ' %! ="& && d & c. s & . TG +& & ="&

K

@;N y = f (x) U& + i: !

G d Tf (x ) > S T24 ^ =%! ]KN x Tf (x) < S < %! 54 x = x @;N y = f (x) =%! Q<_ x = x @;N y = f (x) G d x =

@L& & y = f (x) U& + i: ! ǱL & S T24 ^ =%! ]KN (a; b) @L& & < !+H [a; b] & y = f (x) U& G d Tf (x) > x ∈ (a; b) P

G d Tf (x) < x ∈ (a; b) P ǱL & S < %! 54 ="& Q<_ [a; b] & y = f (x) U& [a; b]

QZ

[a; b]

mH =a

< cos a < < c < a < π ^&& T < c < a Q< π " i: %! : = a < √ + =cos c sin a =x = a

;N ' y = f (x) U& M %fC 8 ^ & Q & %D M+/ 8" T%! $ & VP LP = V+! TV+ " fX d G !+H %Q ! L 8 G d Tf (x ) < S L = B< =P %! ^8. 9=9 8"

46

π

TV+ ! [; a] @L& < %P f (c) a − c f (a) − f () = = = sin a − g(a) − g() g (c) cos c

∈ (; a)

TO& @+A/ (a; b) L& @L& & TW&& ! T(−∞; b) T[a; b) T(a; b] / G @L& P

@L& & y = f (x) U& + i: " ! 24 ^ ="& ]KN (a; b) @L& & < !+H F6

345 U1#5 7 M*N . / 0*1 ,

KL #4 7G $ M*N

" 4 x = x L G.P ' y = f (x) S w L G.P ' y = f (x) G d T"& %De y = f (x) < =%! 54 ; x = x " 4 x = x L G.P ' y = f (x) S w} L G.P ' y = f (x) G d T"& y = f (x) < =" B 6=9 8" & =%! Q<_ ; x = x

G d Tf (x)

G d Tf (x)

= x + x

U&

2

f (x)

!

2 N

=V+ V+! f (x) = x − U& ?+P y = f (x) < Df = R − {} " B + =f (x) = x + +
lim {f (x) − (x − )} = lim

x→∞

x→∞

x−

=

="& y = f (x) U& . { y = x + T5 f (x)

= =

x

=

x

U& + ( ! =%! 5N N < 3 N

{7; & B & FG/ 2O/ ) >) # ! @;N < @D M y = f (x) U& S T ]Qr: %! ^8. L 2O L 8 < ' G d T"& x = x P y = f (x) G d Tf (x ) > < f (x ) > S w[Q =%! 5N < 54 x = x y = f (x) G d Tf (x ) < < f (x ) > S w =%! 3 < 54 x = x y = f (x) G d Tf (x ) = < f (x ) > S w =%! x = x 54 [;C @;N y = f (x) G d Tf (x ) > < f (x ) < S w =%! 5N < Q<_ x = x y = f (x) G d Tf (x ) < < f (x ) < S wP =%! 3 < Q<_ x = x y = f (x) G d Tf (x ) = < f (x ) < S w< =%! x = x Q<_ [;C @;N y = f (x) G d Tf (x ) > < f (x ) = S wL =%! x = x 5 .++ y = f (x) G d Tf (x ) < < f (x ) = S wn =%! x = x 5 ._ G & U& 7 : & B< _+ G Z %Q ! T" 4 x = x L G.P ' y = f (x) S w =%! %&E x = x L G.P ' y = f (x) G d

U& . w[Q F=9 8" f (x) = xx−+ U& . w x +

mH =f (x) =
=V+ %! 2DC y = f (x) @ < Df = R " B ]Q < x(x + ) = + Tx + x = L Tx = L mH =%! %De .P f (x) = x + T
f (x) = x + x

<

P ǱL & S w[Q =%! 5N [a; b] & y = f (x) ∈ (a; b) P ǱL & S w =%! 3 [a; b] & y = f (x)

x ∈ (a; b)

ǱL & < "& !+H [a; b] & y = f (x) S ! T%! %&E [a; b] & y = f (x) G d =f (x) = Tx ∈ (a; b) P =f (x) = c Tx ∈ [a; b] P ǱL & B< c 5 Tx ∈ (a; b) P ǱL & < "& !+H [a; b] & y = f (x) S < c 5 T%! ; [a; b] & y = f (x) G d =f (x) = =f (x) = cx + d Tx ∈ [a; b] P ǱL & B< d

U& 5 : 6=9 8" V! f (x)

>

(x)(x − ) − ( )(x + ) x − x − = (x − ) (x − ) √ √ (x − ( + ))(x − ( − )) (x − )

FF

@$P8 $ L 5 E F 7 M*N . / 0*1 2

KL #4 7G $ M*N

^ L | < %De

=. V+! I=9 8" T^&& <

f (x)

−

! & L U& L ' P . ! + V+! TO& " n" \< L ) f (x) = x − x ,

+x −x

)

x− f (x) = , x +

f (x) =

,

)

f (x) =

)

f (x) = x − tan x,

)

f (x) = e x − x ,

)

f (x) = sin x +

x

e , +x

)

f (x) =

)

f (x) = xx ,

)

x

) f (x) = arcsin

+x

f (x) =

< %De (

; ∞)

& y = f (x) TmH V .

√

+

− + √ − − \ K' ( − ) −∞ " $ +∞

√

+∞ + √ + ( + ) x +

2 N

⎡

sin(x)

x≥ ⎢ −x ≥ ⇔ ⎢ ⎣ x≤ −x ≤ ⎡ x≥ ⎢ |x| ≤ ≤x≤ ⎢ ⇔ ⎣ ⇔ x≤− x≤ |x| ≥

x( − x ) ≥ ,

x , ( + x)( − x )

.

$ TDf = (−∞; − ] ∪ [; ] ^&& T$ @ !+H x = − < x = mH T%! N

f (x) = ( + x)/x cos(x), ) f (x) = (x − )(x − )(x − ),

)

1

f (x)

=

=#N7

f (x)

=

# M4 BC 6 J(K - ./(0

2

−x

+

−

−x

√

√

x − x − (x( − x )) √ + ⇔x =

√

+ x = ± + T^&& {+ & TU& ^ V+! & V N mH =%! ,D/ L DC −

P L & + %&E " =arcsin x + arccos x = π/ [− ; ] f (x) = arcsin x + arccos x U& p ^ & = x ∈ (− ; ) P L & 24 ^ =V+S p

x = −

& =. 2DE M '. & +& P3 & 5& M S " ! %+4 ^ L p ^ =%! %&E p @L& & U& d T"& 4 L& '

& y = f (x) T+ =f (x) =

−

) &

=V+ V! f (x) = x( − x ) U& S < S x ∈ Df " B

) f (x) = x − x + ,

x ∈

^+& y = f (x) TmH ="& %De L&

=. V+! `F=9 8" ^&& <

cos x , cos(x)

)

√

x −∞ f (x) − f (x) + f (x) x +

U& . I=9 8"

+

(x − )

=%! (−∞;

f (x) = x( − x )/

<

(x − )(x − ) − (x − )(x − x − ) (x − )

= =

√

, " p f 2CfX ^ .P
+ ,

x −∞ − f (x) + f (x) − f (x)

=

FI

√

+

√

,

− ,

− −

−

√

√

− / − − +

− −

KL #4 7G $ M*N

@$P8 $ L 5 E F 7 M*N . / 0*1 2

2 N

P L & T^&& < f () = π/ =%! %&E (− ; ) @L&

" V8 /& =%! " 2DE V8 x ∈ (− ; ) =. N+N3 k.+N x = < x = − L & %&E cos x + sin x = ( + cos(x)) 3 =+

U& p ^ & =

=x− x < ln( +x) x > P ǱL & + %&E p f (x) = ln( + x)− x+ x U& p ^ & = T24 ^ = < x V+ i: ^+t.P TV+S V Tx > $ f (x) =

−

+x

2 N

x +x

+x=

Tx > S mH =%! 54 [; +∞) & y = f (x) T^&& =%! p < .P Tf () < f (x) G d

G d T < y T < α < β S + %&E =(xα + yα )/α < (xβ + yβ )/β P(Q y < β Tα V+ i: T< ^ 2DE & = < ! < $ T
f (x) = cos x + sin x −

( + cos(x)) 24 ^ =V+S p R &

2 N

α

ln(xα + y α ) <

β

f (x)

f (x) >

xα

α

ln(xα + y α ) −

β

ln(xβ + y β )

^&& =f (x) =

xα xβ > β α +y x + yβ

xα− xα +y α

)

⇔

xα (xβ + y β ) > xβ (xα + y α )

⇔

xα y β > xβ y α y β−α > x y > ⇔ y>x x

⇔ β>α

⇔

f (y) = lim f (x) =

x→+∞

ln(y β ) − ln(y β ) = , β ln − > α β α

α

α

=

α

lim ln(xβ + y β )

lim ln(xα ) −

β

lim ln(xβ ) =

x→+∞

≤ f (x) ≤ ln

α

−

2

P ǱL & + %&E " √ =%! /& x > − < x

x

V+ i: Tp ^ & = 24 ^ = < x <

="& i: T Q<_ ( ; +∞) @L& & y = f (x) U& ]Q < f (x) < ^&& 5 Tf (x) < f ( ) x > P ǱL & + =%! √ √ = − < x − x + − <

V x > P ǱL & T^&&

"

√ − −x x + f (x) = √ + = x x x √ √ < x x ]Q < < x 5 & < x

x→+∞

x→+∞

+ +− =

f (x)

x + cos(x) =

√ f (x) = − x + −

α

lim ln(x + y ) β

sin

x >

x→+∞

−

− sin(x) cos(x) + sin(x) =

U& & Q<_ < 54 @5Q; & +& < U& ( T G 74 8 =. 2DE =%! {!

x→+

lim f (x) =

=

cos x = sin x + ( − cos(x)) ) arccos − x = arctan x ≤ x +x ) arcsin x = +x ⎧ x ≤ − S ⎨ −π − arctan x arctan x − < x < S = ⎩ π − arctan x ≤ x S

Ty < x S < < f (x) G d T < x < y S T5 Z P%Q %! : mH =%! f (x) < G d lim f (x) < f (x) T lim f (x) V+G& p x→+∞ x→+

− sin x cos x(sin x − cos x) + sin(x)

)

xβ− xβ +y β

24 ^ 5 =%! xf (x) > 5 & −

=

+ 2DE L P3 L ' P

(x > )

ln(xβ + y β ),

− sin x cos x + cos x sin x + sin(x)

:X L =%! %&E R = (−∞; ∞) @L& & =" 2DE V8 < f () =

V+ [5 L U& T+Q ^+.P & f (x) =

=

β

= %&E P& ,< <

x

F1

x

@$P8 $ L 5 E F 7 M*N . / 0*1 2

KL #4 7G $ M*N

Tx Tx C n P < n ∈ N P ǱL & TO& @+A/ M&; TmH V I @L& U/< xn < . . . ln

+ %&E

=ex > G d Tx > S w

x + x + · · · + xn ≥ n ln(x ) + ln(x ) + · · · + ln(xn ) ≥ n ≥ ln (x x · · · xn )/n

+

x

+ ··· +

xn

≥n

=

=

− /x − /x − · · · − /xn n

G d Tx = S w9

+ ln x ≤ x

G d T < x S wF

+ x ) x P ǱL &

<e<

x

x P ǱL & w1

< sin x < x G d T < x <

x

+

x+ +

x

> α(x − ) G d

T

wI

π

S wJ

G d T < x S w ) <x

<

√ √ √ n x − n a ≤ n x − a G d Tx > a >

< α S

w

∗

< n ∈ Z S w 0

wd < M & T5x U& & 3 L + B L @+A/ & =. ! < 2DE " < "& 3 I @L& & y = f (x) S

(Q C n P < n 5+DX C P ǱL & G d T (Q n ∈ Z V x , x , · · · , xn ∈ I

=d %!& ] < n ^+:X & <

f (x ) + f (x ) + · · · + f (xn ) ≤f n

" "

< . . . Tx Tx < n > Tm, n ∈ N S + %&E w 24 ^ T"& P(Q %De C

x + x + · · · + xn n

E yS U&z '+!f L ) @34 {7; ^ 2DEx w=%! d ^+d +

C f < & <" 2

m m nm− (xm + x + · · · + xn ) ≥ (x + x + · · · + xn )

m

%&E Tx %De < N+N C n P < n

m m m− (xm ≥ (x + x + · · · + xn ) + x + · · · + xn ) n

m

, · · · , xn ∈ R < n ∈ N

=x − x

=xα −

x + x + · · · + xn /x + /x + · · · + /xn . ≥ n n

G d Tx , x

x

+

G d Tx > S w>

+ x ln x + +x ≥ +x

=

T^&&

xn

x+

=x arctan x ≥ ln(

(; +∞) @L& & y = U& T ^ 2DE & f (x) = G d T < x S 24 ^ =V+S p xn < = = = x Tx %De C ǱL & ^&& =−x− < V ≥

)

=

− x

n − x +x +···+x n

(x −

G d Tx > S w0

=|x| ≥ | sin x| x P ǱL & w6

x

)

=cosh x >

2 N

(x + x + · · · + xn )

=x > ln(x + =ln x >

! e & & T%! 54 x → ex U& 8 & B & < =S + ] < T^+:X Tx %De C n P < n ∈ N P ǱL & + %&E xn < . . . Tx

"

∈ N

! "

P ǱL & T5 =+

V xn < . . . Tx

√ n n x x · · · xn ≤ x + x + · · · + xn .

S + %&E w0

@L& & f (x) = ln x U& L T< ^ 2DE & M L T%! 3 f (x) =V+ ! (; +∞) %! I & d < I =

ex + ex + · · · + exn ≥ n e(x +x +···+xn )/n n ex + ex + · · · + exn ≥ n exp x +x +···+x n

f (x) = (f (x)) =

FJ

x

=

− x

?) #: <LK+4 $ < #4 7G P 7 M*N #4 7G I

KL #4 7G $ M*N

2 N

x

# 9!36F +N 6 J(K !36F

V+ 3 +; + y = A+& a b % ^ +& < A+& P3 L d f

="& %.! U/< .C U7 & 3 V+ i: `1=9 8" T^&& =%! x && P x 3 & T%! ^ =y = ±b − x /a + =. V+! A(x) = x × & %! && 4 +; % T24 ^ T = ≤ x ≤ a V B _+ < b − x /a 24

− x

x + bx a a − x x ⇔ − = − a a √ a ⇔ x = ⇒x=a

A (x) = ⇔

b

−

x a

<' !D 9MT)3 =" ! 54 < & L +& M & =V+ h d L ! < Z . $ u ^ {Q/ = : 2CfX & L+ B Qe %! D h & T d & 24 L mH Th = "

2d

>) U7 & < U& 8" & N @5;/ '$ U& $ & & V+P =%! + ! ^ =V+ + L& < +" @D5B TU& ^ ! $ L =" e ] @D5B V V+P G$

B 5& U7 ,X V+ i: Tp ^ & `1=9 8" &x "& x && i< U& @"S $ L " V (x) = x × ( − && 4 @D5B V =w" B [Q mH = ≤ x ≤ V B ^+t.P =& P x) ^ & =V+ [; ] @L& & V (x) U& %! :

V+ D!3 V (x) p

=

8 & B & =%! ] @L& & U& ^+8 @;N

√ = ab TA() = V+S + A(a) = < A a x = ǱL & %! ab && L& ^ A(x) N e

√ =P a /

V (x) = ( − x) − x( − x)

2 N

5" & < + ' O& k.+N & ! : d & C $ V+ & V+P =" _
%! 5 ^ & V (x) = T^&&

" 3 +; w

R R = l x +R

k C + TA(x)

≈

_+ < " %: $

/l =

/(x + R )

V (x) = ⇔

R

⇔

A(x) = k × × x +R x +R

=V+ A (x) =

≤ x

i: &

⇔ kR ⇔

−

x=

A(x)

H

+" @D5B w[Q A+&

− x = − x − x = x= x=

="& ] @L& U& ^ ^+8 @;N x = T+ N V ( ) < V () TV () N %! : ^&& && [; ] @L& & V (x) N e T@+ =V+

4 x = ǱL & %! {58 ! V () = ="

U& %! : mH p ^ &

(x)(x + R )−/

I)

"*;L /D J

KL #4 7G $ M*N

_+ T+& ^8. @;N ^^+hH & UD S 5 =" B J=9 8" & =& P e d (&

d %C! {58 & (& ' %! vj wF & 7+ ) %C! V+ =%! {! |( < %! %C! F) %! %.+/ T%C! %C! 91) & Q& w%C! L N x G P |( . L C! $ =" |( . -& P ^ & ! L 7+

-& P N$ %C! P . + V! $ ! s P = ( , ) @;N L wI B 2j ( P3 < De 25;/ ="& ^ .

d < "< + < c : J=9 8"

d L H H < R @C/ 5" & X<( [+/ ' w1 5" = : [+/ ^ ^+G! Q7S ' =%! " & Td ^8. V ^ S_& "& & N$ Q7S ^ | D [+/ L TQ7S ^ @" <Xl W(& {D! -"

( EM . A

I

^++5 " & U& V %./ L S ^ =P & %Q 8 & < T

P\< ^ =" ! D < L VP \< & "< !H <& h L 5+!< @ ! < P %<3 _+ " @+A/ L N; + , +HP h/ = P =%! 7 ^ !H T%! ∞ Z[ ! lim f (x) 9" : x→x g(x) ∞

√ f ( a) =

lim

f (x) g(x)

= =

lim

x→x

f (x)−f (x ) x−x x→x g(x)−g(x ) x−x

V

+& $ C < . 24 & 1 C w ="& ^8. N ^ 8$ d 2D58 . 8 d TS i< % & P+; U+.B L w0 =+ ^++5 %! s+3 ^ .

d @h/ +G& p hNQ 7e w> U7 ,X =%! V V T%! fAQ < N ^ . d % "& & N$ d @C/ -"& ^8.

f (x ) g (x )

∞ ∞

%Q

2

− gg(x) f (x) g(x) () (x) = lim = lim = lim x→x g(x) x→x x→x − f (x) f (x) f (x) g (x) f (x) g (x) × lim = lim = . lim x→x f (x) x→x g(x) x→x f (x)

%Q L w x 8 Y+ =

3 V ^ +& & ! TR 5" & w9 =+

& ! 0) ,X & Q & X<( [+/ ' w6 "& & N$ <( ^ =" ! -" 4 ^8. V ^ S_&

f (x) g (x)

^&& x→x =V . !

= lim

= √ > a

=

x x=√a

√

f (x) − f (x ) g(x) − g(x )

lim

a

5 =%! y = f (x) 5 .++ ' x = a mH = √ N / x = y = a ǱL & p . √ =%! a && N ^ < "&

x→x

x

(x) /4 !4" <= . // lim fg (x) 4 1 @ " 4 4 " V %Q A7S" x→x

<

2 N

%De C < . N ^ .

=V+ ^++5 T%! a %&E && d A74 p U.B 4 U& =y = a/x ^&& Txy = a $ f (x) = " 24 ^ =f (x) = x + a/x V+S √ x = ± a x = a + T%! − a/x = 5 & √ ^+8 @;N x = a mH T%! " i: x > $

="& 7 y

I

KL #4 7G $ M*N

ln(cos(ax)) , x→ ln(cos(bx)) x ) lim xx − ,

)

)

lim

)

ex − − x→ cos x + lim

x

−x−

x

−

xx − x x→+ ln x − x +

,

lim (tan x)tan(

x)

lim

)

x→+

x

7)5 CD V

x→π/

!z 5 & wPx . TL DC =a > < "& y, +HP \< L ,

,

( + x)/x /x ( + x)/x − e , ) lim x e x→ x→+ lim (a + x)(b + x)(c + x) − x , ) x→∞ # ) lim x + x + x +

)

ln( + x) P /( + x) = lim = x x→ xm − am P mxm− m lim n = lim = am−n n x→a x − a x→a nxn− n ax P ax ln a P ax (ln a)n P = +∞ lim n = lim = · · · = lim x→∞ x x→∞ nxn− x→∞ n! x VD %Q lim − D!3 & ln x x→ x − V+ .C L \< & T ∞ − ∞ lim

x→

lim

2 N

x→∞

−

x +x+

lim

$

×

x→

ln(ex + x) . x

x ln x − x + = lim ln x x→ (x − ) ln x ln x + x /x − x ln x P = lim = lim x→ ln x + (x − ) /x x→ x ln x + x −

x x−

P

−

= lim

6' 4 @A

U

x→

Tln x

ln =

···+

n!

f

e − , x→ sin x x − arctan x lim , x→ x /x arctan x lim , x x→ lim arccotx − , x x→ , lim x e/x −

) ) )

= f (x ) + f (x )(x − x )

)

(x )(x − x ) + O (x − x )

n

x→

P

= lim

sin x

x→

x − cos x sin x

− sin x × lim lim × lim x = x→ cos x x→ x x→

) lim

(n)

x→

x

) /: #

n

ln x

lim sin x ln x = lim

%!& L < L ' P N

3: y = f (x) 2+ 9" N" (n − ) + + 7%Q "" x = x ," <:? A<= . % " /4 f (x) 4

(n)

x→

= = ln = ^&&

x→

=%! " D × ∞ VD %Q &

& =f (x) = g(x) + O (x − x )n V+ %Q ^ =" ! _+ f (x) ≈ g(x) j . L U/

+ f (x )(x − x ) + · · ·

2+ N

lim xsin x

x→

x→

=

f (x)

ln = ln( lim xsin x ) = lim ln(xsin x ) = lim sin x ln x

< y = f (x) + i: # < S [5 x = x @;N G.P "& 5& x = x y = g(x) < y = f (x) V+hS 4 =n ∈ N f (x) − g(x) =x→x lim = P n @D & n

=

D!3 & V+ .C L \< & T G !+H +Q & mH T = lim xsin x V+ i: x→ V

y = g(x)

(x − x )

ln x + x x +

VD %Q &

" n; ,! ^ T P 7 ( U& - ! ! U& L +t+H U& & d ^ ! 7.B $ 8 & B & =DN 8" & / U& d V+ n; ,! ^ T P U& & x = x i< @;N G.P y = f (x) i< ^ & T%! ^+$ S -L {N n @D L 7.B $ ' -%! ^8. {N

ln x + x x

x→∞

− ln x

)

@ " I:%[ /0 \/ ]:+ I:" 4

lim

x→+

x , (− ln x)x

x ) lim x/ ln(e −) , x→+

I0

x − sin x , x→ x − arctan x ax − b x ) lim x x , x→ c − d

)

) ) ) ) )

lim

lim xn e−/x ,

x→

lim

x→

ln x

−

x ln x

,

lim x e/x ,

x→

lim (tan x)

x−π

x→π/ −

arctan x

lim

x→+

x

,

,

7)5 CD V

@;N

KL #4 7G $ M*N

VP @D f (x)

=

sin x

V" @D f (x) = ln

2+ N" n + /$+ " O K/ 6 " 7 #? " 7 I:" .x = 9" @Z x = x y = f (x) @Z G/! 3

U& ! @D 7+ s& w0 =
f (x) = xex

sin x x

U& Q ' s& w> =

2

U& $ @D 7+ s& # √ =+&+& x = f (x) = + x y = f (x) $ @D 2N Tp ^ & = V+ D!3 x =

U& Q ' s& w9 =
+ 2DE L < L ' P TQ ' s& '. &

f( ) =

xn ) ex = + x + + x + · · · + + O xn ! n! x x ) cos x = − + − · · · ! ! x n · · · + (− )n + O x n+ (n)! ) ( + x)m = + mx + m(m − ) x + · · · x

m(m − ) · · · (m − n + ) n x + O xn n! + x + x + · · · + xn + O xn

···+

)

)

−x

=

f ( )

ln( + x) = x −

x

+

f (x)

(sin x) − (−x) x x + − · · · ( − x + x − · · ·) x− ! ! x

x

x

=

x−

=

x + · · · − x + · · · ! x + Ox x − x + x − x −

!

+

!

− ··· − x +

!

(Q |

(x − ) +

−

(x − )

× − + (x − ) + (x − × ×

f () =

f () () = · · · = f (n) () =

f () =

f () () = · · · = f (n+) () =

f () =

f () () = · · · = f (n+ ) () =

f () () =

f () () = · · · = f (n+) () = −

− ···

sin x

)

=

+x+− x !

++

x x +− + ··· ! !

x x x + − + ··· ! ! ! x n+ · · · + (− )n +O x (n + )!

= x−

2 N

U& Q ' s& =
x=

^&&

· · · + x −

+

+

2 N

2

=

x=

f (x) = sin x U& (Q @D Q ' s& =
U& Q ' s& # sin x =+ D!3 VH @D x+ >= )=9 L w>x ,e < 9= )=9 w1x < w x ^. L = V+S + < ! =

=

+O (x − )

L & < ! d %!& 2CfX L =. ! 24 &

sin x x+

=

n

f (x) =

− / ( + x)

− ··· x + O xn · · · + (− )n− n

x=

= − − f ( ) = ( + x)−/ = x= ( + x)−/ = f () ( ) = x= − − f ( ) ( ) = ( + x)−/ =

x

=

( + x)/

n+

2

@;& & U& ! @D 7+ s& # =
+ x = ( + x )/

x−

I>

KL #4 7G $ M*N

P8#0

2

U& Q ' s& L ! &

lim

x→

cos x − e x

=

= # −

−

x

lim

=

x −

x

x

−

+

x

+

x→

V Te < cos x

+

x→ x

=

+

=

x

−x /

lim

−

#

x

+ O x

+ O x

x

=

+ +

x

−

x

+

− ···

− − n! −(n − ··· •

•

•

+

=

+ O(x )

x

x

−

+

x

··· ) x

•

lim

x→

x

+x+

x

× x− =

lim

x→

x

x

−

x

) lim

x→

) )

lim

x→

x

−

sin x

x

+

x

sin(sin x) − x x

+ O x

lim

+ x −

=+&+& f (x) = x

× $

+ O x

− x( + x) =

−=

x→

x ex −

n

+ O(x

n

)

x→

x

+ x

−x

−

f (x) =

f (x) = sin(sin x)

)

−x

) )

U& ! @D Q ' s& w> =+&+&

+ 2DE L DN DC L ' P x ≈

−

R

VP @D Q ' s& w

U& VP_+! @D Q ' s& w0 =+&+&

#

− (cos x)sin x x

lim

)

2 N

+ O x

)

n

! #

+ D!3 L < L ' P

+ O(x

− ···

(− )n (n − )! · · · + n− x (n − )!

lim

=

n

•

V Tex < sin x U& Q ' s& L ! & ex sin x − x( + x) = x→ x #

x

•

$

−

=

(

( /)( / − ) x + (x ) /)( / − )( / − ) (x ) + · · ·

(R + x) R +x −x − ≈ −x +x

ln ln( + x/

)

≈

x

x

+Q = ! < @D!3 7+ s& L =V d L %./ fkD/ %! L +A/

+ i: @;N y = f (x) U& V+hS 24 ^ T%! ]KN x = x @;N d + < %! ]H + x = x TV+ [5 df |x = f (x )dx 24 &

S (I ) 6 ( S/> 4/ ? ) " # ' y = f (x) U& (n − ) @D h_B 2N " B< _+ f (n) (x ) < "& B x = x L G.P G d T"&

+ N7!/ x = x

y = f (x)

#

dx = x = x − x

f (x) −

%! +a < 5& df ^&&

f () + f ()x +

···+

df : (x, x ) → f (x )(x − x )

I9

(n − )!

f

(n−)

f ()x + · · · $ n−

()x

∼

n!

f n ()xn

- . / 0*1

Tx

KL #4 7G $ M*N

< f (x) = cos x V+ i: ! x @;N f k @D s& y = gk (x) S 24 ^

G d T"&

U& . & c. s S 2456 ; d < . V! x = x @;N y = f (x) V+! y = f (x) & DN N ' & TV+ % c. s & 54 _+ df |x T5 =f (x) ≈ f (x ) + df |x = ++a Δx @L & x L +a GP T%! U& & =" B [Q` )=9 8" &

=

V+!

g (x)

=

g (x)

=

g (x) =

−

x

g (x)

=

g (x) =

−

x

≤ k ≤

,

&

gk

<

f

, +

x

,

U& ` )=9 8" =V .

U& k @D 7+ s& P " L G.P ' d < : x = x @;N y = f (x) + V+! x = x ) f (x) = x sin x,

x = ,

k = .

)

f (x) = x − x + ,

x = ,

k = .

)

f (x) = x − sin x,

x = ,

k = .

)

f (x) =

x− x+

x = ,

k = .

,

+, - ./(0

@D ;& w + !P +D5 w[Q i< U& ' O& ]KN y

= g(x)

= f (x)

S

24 ^ T"&

)

) d(af ) = adf,

)

d(f g) = gdf + f dg,

)

d(f (g)) = f (g)dg.

d(f ± g) = df ± dg, f ) d (gdf − f dg), = g g

= V+.5 L 8" & +

@D M y = f (x) U& S & < dn f . & d n @D + T"& L ! & =dn f = f (n)(x)dxn V+ [5 L 24 %" ! 24 & 7+ @+A/ T ^ n

%3 W(& & TK+ _: L ! 2N @P & =" 5B ' j: L ^+.P i: X B FG ^ Y* ; A =V . [5 K+ s+3 fkD/ y = f (x) U& +

! diff(f(x),x) ! L U& ^ M @D!3 & =V+ ! L f (x) U& n M @D!3 & =V+ ! diff(f(x),x$n) . sN: TV+ ! Diff L diff & t $ P( 24 D!3 < " P 4 +GN =%:]H + B L n" & Z $ &

' y = f (x) S ) /: &4U C G f (n) (x ) < "& " n @D M x = x L G.P

G d T"& B f (x) =

f (x ) + df |x + +

@;N y

n!

df

x

+ ···

df n |x + O ((x − x )n )

U& k @D_ 7+ s& g(x) t $ = g(x) U& . L %! 2DC dk f G d T"& x = x = f (x)

K+

y

K+

" B = g (x ) dx TG +& & =x = x @;N x %Q T%! %! s ' y = g(x) Tk = %Q

= = = < %! .! ' y = g(x) Tk =

diﬀ(xˆ2 − 3 ∗ x + 1, x) −→ 2x − 3 diﬀ(sin(1 − 2 ∗ x), x$3) −→ 8 cos(1 − 2x)

K+

Diﬀ(sin(1 − 2 ∗ x), x$3) −→

∂3 sin(1 − 2x) ∂x3

I6

=Δg

x

KL #4 7G $ M*N

- . / 0*1

& y = f (x) U& .++ @D!3 & & 24 &

! minimize(f(x),{x},{x=a..b}) ! L [a; b] @L& & y = x − x U& .++ < ._ T . & =V+ L DC {+ & [− ; ] @L&

U& S 25O FG Y*

" :5 . 8" & F (x, y) = c @;& '. & ! L x { & y n @D M @D!3 & T"& & =V+ ! implicitdiff(F(x,y)=c,y,x$n) %D y ! M 24 ^ Tx + y = S T . & %! && x & K+ x(x2 + y2 ) y = f (x)

K+

maximize(xˆ2 − 3x, {x}, {x = −1..1}) −→ 4

K+

minimize(xˆ2 − 3x, {x}, {x = −1..1}) −→ −2

implicitdiﬀ(xˆ2 + yˆ2 = 1, y, x$3) −→ −3

s& @D!3 & FG ^ ) /: 9$G ! ! L x = a @;N f (x) i< U& n @D 7+ s& T . & =V+ ! taylor(f(x),x=a,n+1) L %! 2DC x = @;N /x U& < @D 7+

U& S ( ) J FG Y* H 8" & y = V (t) < x = U (t) s&< '. & L x { & y n @D M @D!3 & T"& " :5 implicitdiff({x=U(t),y=V(x)},{y,t},y,x$ n) ! Ty = t − < x = sin(t) S T . & =V+ ! & %! && x & %D y ! M 24 ^ y = f (x)

K+

taylor(1/x, x = 1, 3) −→ 1 − (x − 1) + (x − 1)2 + O(x3 )

%! 5 ^ & K+ O

(x − x )n

. " B

implicitdiﬀ({x = sin(t), y = tˆ2 − 1}, {y, t}, y, x$3)

K+ sin t cos t + t cos −→

O (x − x )n−

cd

y5

t + sin t

cos t

._ @D!3 & "

http://webpages.iust.ac.ir/m_nadjafikhah/r1.html

=%! "
FG ./$'

! L [a; b] @L& & y = f (x)

maximize(f(x),{x},{x=a..b})

=V+ !

IF

!" #$ "$

P L & B< C &E C =F (x) = F (x) + C =x ∈ (a; b) Tg(x) = F (x)−F (x) V+ i: (a; b) & _+ g(x) TF (x) < F (x) ]KN +Q& 24 ^ x ∈ (a; b) P ǱL & i: M&; T
%! Q@ ^ j: ^ L vP

x ∈ (a; b)

`b)-!

g (x)

=

F (x) − F (x)

=

f (x) − f (x) =

5 & d T " & i< y = f (x) U & Sz =yF (x) = f (x) %: y = F (x) @ p z & & < {QB +& @ p ' Lld Q@ ^ d <& 5& j: 8 / =%! y+ 2O5 = . $

=%! %&E (a; b) & y = f (x) mH

2

5!4

U& & +Q< U& ' y = f (x) S U- @+Q< U& . @C. G d T"& (a; b) @L& & y = f (x) ={F (x) + C | C ∈ R} L %! 2DC (a; b) & y = f (x)

=(a; b) ⊆ Df < y = f (x) + i: FG (a; b) y = f (x) 4 y = F (x) U& =F (x) = f (x) x ∈ (a; b) P L & V+hS

U& & +Q< U& ' y = F (x) S &&) U& @.P @C. 24 ^ T"& (a; b) @L& & y = f (x) & {F (x) + C | C ∈ R} & (a; b) @L& & y = f (x) @+Q< V+ 5 =V+P F (x) + C .

& f (x) =

x

<

(−∞; ) ∪ (; ∞)

F (x) =

/x + /x

x> x<

x> x<

) @L&

2

& f (x) =

−x

U& &

2 N =%! R = (−∞; +∞) & /x U& & +Q< U& ' F (x) = ln x 2 N 2 N

2+ N

S S

5 ^ & T%! + N %+4 U& P M =%:S P _+ d ^+& N @.P T+G& < M S =+ / N Q< +S < N [x] Q

@C. & f (x) = ln |x| U& +Q< U& Q T P

F (x) − F (x) =

−x

=%! (; +∞) @L& x − x + U& & +Q< U& ' F (x) = ln x +x+ =%! R & f (x) = x − x +x + $ T +Q< U& ?+P R & f (x) = [x] U&

O& + (a; b) @L& & $K8 " ! ,e C & TL =%! _Q F (x) =

−x f (x) = cos x U& & +Q< U& ' F (x) = − sin x

(a < x < b)

&! '( !

+Q< U& ' F (x) = =%! (− ;

f (x) dx = F (x) + C

%

S S

< y = F (x) < y =

G d T"& (a; b) @L& & II

F (x)

S

y = f (x)

U& & +Q< U&

<#:Q#:*8 QP '

W 8 "#:*8 '

) ) )

-

)

cos xdx = sin x + C, -

f (x) dx

π x = k , k ∈ Z

x csc xdx = ln tan + C, (x = kπ, k ∈ Z) sec xdx = tan x + C, x = k

-

π

, k∈Z

)

csc xdx = − cot x + C,

)

x = kπ , k ∈ Z

-

)

x dx + C, ) = arctan a a x +a aπ a = , x = k , k∈Z x − a dx + C, ln ) = a x + a x −a

) )

)

)

-

)

x dx + C, = arcsin a a −x a = , −a < x < a

- x x + a dx = x +a

a + ln x + x + a + C, - x ) x − a dx = x −a

+

)

a

-

arcsin

a

n dx √ = n x n−

+ C,

-

x

x + C,

dx = ln x + C,

(x = )

[x]dx = ∅,

−x

-

dx = arcsin x + C,

(− < x < ).

%

- E- 4K

xa+ + C, − = a ∈ R a+ dx = ln |x| + C, x = ) x x a> ) ax dx = a + C, ln a ) ex dx = ex + C, )

a = , |a| ≤ |x|

√ + C, n x−n

-

V T+GN &

a − ln x + x − a + C, |a| ≤ |x| - x ) a − x dx = a −x x

x dx =

L .P & !@ =" n; M Q@ D/ W(& M: Ok.5 < %(! +GQG T%! M: +GN

+ 2O5 L . Q@ ^ L %! L +.X 24 +Q ^+.P & T(y = f (x)) %! 8 & Tfke "& ,3 %! ^8. d ^ : TB B< U& ?+P Q< T%! +Q< U& y = sin x/x U& V+ =V+!". " sin x/x && d M N =%! M & %SL& \< & +GQG Q@ ,G & UB CfX < ! 7D/ 2CfX L T5 H Q: L ,<B ' nX & ^ =V
dx = ln x + x − a + C, x −a |a| < |x|

-

%Q! &

9! DO!D(7 ON

(a = , x = ±a) = ln x + x + a + C

dx x +a

(a < x < b)

! "! I0(!

-

"

u L T"& ^8. @C. ^ S_& (a; b) S =V+P . = f (x) dx V+ < d %! ^"< =V+ ω = f (x) dx 2DC

G d T"& y = f (x) & +Q< U& ' y = F (x) S

=" && ω & F (x) +

-

-

π x sec xdx = ln tan + + C,

)

! "!

-

)

=(a; b) ⊆ Df < y = f (x) V+S - E- " (a; b) @L& & y = f (x) U& @+Q< U& @.P @C. . & < + (a; b) y = f (x)

sin xdx = − cos x + C,

<x

I1

xa dx =

<#:Q#:*8 QP '

W 8 "#:*8 '

√ √ x+ + x− √ √ √ √ dx x+ − x− x+ + x− √ √ x+ + x− dx (x + ) − (x − ) - x + + x − dx (x + ) + (x − ) + C

= = = =

G d x = − S

)

)

2 N

- x dx = x −x+ − dx x+ x+ dx = (x − x + ) dx − x+ x x = − + x − ln |x + | + C

-

G d x = kπ + π/ S -

-

tan xdx

= =

sin x dx cos x − cos x sin x

-

=

tan x − x + C

x P L & − cos(x) dx

-

sin xdx

= =

x−

sin(x) + C

G d kπ − π/ < x < kπ + π/ S -

-

= = = = =

-

= =

cos x

+C

x P L &

-

x dx + cos x dx = cos x √ dx sgn cos x cos x x √ sin +C sgn cos

x dx = − sech− + C, a a x a −x < |x| < |a| , a =

-

x dx = − csch− + C, a a x x +a x = , a =

f (x) dx = F (x) + C

4

G d T"& 4 [Q( C a <

2+ N

-

g(x) dx = G(x) + C

)

(f (x) + g(x)) dx = F (x) + G(x) + C af (x) dx = aF (x) + C -

)

f (ax + b) dx =

a

F (ax + b) + C

2DC %! %.! " B TW&& S U/< =%! " C. ' 7 8" T,< A+B := 24 & d U.74 T"& C. < B < A ' A74 =V+ [5 {a + b | a ∈ A , b ∈ B} [5 αA := {αa | a ∈ A} 24 & C. ' C C u L T,G " +4 =S ^ =S u C d s < " H @7 =L! ! 4 < /& +B

2# N 2% N

V T ]Q r: {Q; & B &

G d x = S

dx − sin x dx = dx + sin x + sin x − sin x − sin x dx − sin x − sin x dx cos x − sin x dx sec xdx + cos x dx − (ln | cos x|) dx + − cos x − ln | cos x| −

-

-

T

dx

=

x dx = sec− + C, a a x x −a |x| > |a| , a =

)

)

(sec x − ) dx sec xdx − dx

=

-

-

x

+ x − x

dx

= =

-

xdx +

2 N

-

-

dx −

dx x

x + x − ln |x| + C

G d x = S

2 N

- - dx = x + x + + dx x+ x x x dx dx + = x dx + xdx + x x x x = + + ln |x| − +C x

2* N

G d x > S

√

IJ

dx x+

−

√ = x−

2 N

W 8 "#:*8 '

#X* #X5 >$ 7 4 <#:Q#:*8 '

! V( ! V4 ;# 6 3 9! DO!D(7 -

f du =

f (x) du(x)

L p

]KN 5& u(x) T =%! 5& u(x) <

f (x) dx = F (x) + C

-

=

-

%

-

,G =%! (Q <

f (x)u (x) dx

= −

24 ^ T"& ]KN V+ B %! :

f (u) du = F (u) + C

`b)-!

-

=f (u) du = u(x)f (u(x)) dx _+ <

,G & _
2

-

du = tan xdx -

=

ex sin( − ex ) dx

=

ln | sec x| + C

)

− ex V+ i: S ^&& Texdx = −du/

2 N

=

−

cos( − ex ) + C

x G d u = tan x V+ i: =

√

S

)

tanh xdx

)

-

x

) )

2 N

)

^&&

)

2+ N

)

) )

du = sin x cos xdx = sin(x)

1)

x+

dx,

sin x sin(x) dx, -

u du = u/ + C

(tan x)/ + C =

G d u = sin x V+ i: S

-

)

(sin x + cos x) dx,

)

cos u + C

,

-

)

sin u du − sin udu =

=

- √ tan x dx cos x

+ <

2_ N

dx √ √ x+ x+

-

)

+C

=

< du = dx/ cos

2 N

udu =

Tdu = −exdx G d u = -

-

=

)

)

u

cos x + C

+ D!3 L QG L ' P

= ln |x + x + | + C

-

tan xdx ln | sec x|

(x + )/ (x + )/ − +C / / (x + ) x + − (x + )x +

=

du = ln |u| + C u

G d u = ln | sec x| V+ i: S

cos(x) +

x x + dx = (x + − ) x + dx - (x + )/ − (x + )/ dx = =

V+ i: S + < du = (x + ) dx G d

x + dx + x +

G d x > − S

{F (u)} = u (x)F (u) = u (x)f (x) 2

sin(x) cos(x) dx = (sin(x + x) + sin(x − x)) dx = = (sin(x) − sin x) dx

&& )

S

2, N

x P L &

-

)

- -

− x dx,

)

−x

)

-

dx,

sin xdx,

- -

) )

e dx,

− sin x dx,

x + x + dx, x+

-

(tan x + cot x) dx,

- √ x− √ dx, x ) ( − x) dx,

dx , x −

x x

-

+C

) )

√ x(x − x + ) dx, sin x − cos(x) dx,

-

xdx √ . x+ dx , x +

x + dx, x+

√ -

a

dx −x

x − √ ax

, dx,

sin xdx, dx sin x cos x

,

#X* #X5 >$ 7 4 <#:Q#:*8 '

24 ^ Tu

=

a tan x b

I

= = = =

W 8 "#:*8 '

V+ i: S

^&& < du =

-

dx ) , x + - arcsin x ) dx, −x ex dx ) , ex − sin(ln |x|) dx,

) x + sin(x) ) dx, cos (x) ) tanh xdx,

) )

)

-

-

-

) ) )

)

sin(x) dx

ex du = eu + C

=

ex

arcsin x + x dx, −x

) )

-

dx cos x +

dx + ex

= dv/

< du

=

= -

=

V+ i: S ^&& < du = ( −

= x + /x

2* N /x ) dx

(x − ) dx (x + x + ) arcsin(x + /x)

= du/(u + ) I

= = =

1

V+ i: S , +

( − /x ) dx (x + /x) + ) arcsin(x + /x) du) dx (u + ) arcsin u

=

< dv

/ = v/

dv/ dv = (v/) − ( /) v − v − + C = ln u − + C = ln v+ u ex +C = ln x e +

I

du du = u(u − ) u −u du (u − /) − ( /)

G d u −

V 24 ^ Tu dx,

|

=

√ sin x cos x dx,

- ) x+

2% N

-

-

-

dx + ex

,

V+ i: S < dx = (u − ) du ^&&

G d u = + ex V+ i: S ^&& < dx = du/(u − ) + =x = ln |u −

=

x.x dx,

2# N

√ x

u

-

-

+

+C

= u−

ex dx

, −e x √ ) tan√( x) dx, x dx , ) sinh x - ) x − x dx,

G d u =

x

− u + u − ln |u| + C √ √ ( + x) − ( + x) √ √ +( + x) − ln | + x| + C

=

ln(x) dx , x ln(x)

-

)

=

dx , − sin x + cos x ) x − arctan(x) dx, x + - x − dx dx, ) , ) x (a + b) + (a − b)x ln(x + x + ) ) dx, x + dx , ) ( + x ) ln(x + +x ) ) tan (x) sec (x) dx.

x

+x + (u − ) √ dx = (u − ) du + x u (u − u + )(u − ) du = u - = u − u + − du u

-

arctan(x/) dx, x +

-

)

Tx = (u −

x+ dx, x −

-

)

ax + b dx, cx + d

esin

^&& <

= esin

dx,

xe -

−x

a du b cos x

a sin x + b cos x dx dx • b cos x (a /b ) tan x + du arctan u + C = ab ab u + a arctan tan x + C ab b

x dx, + x

)

-

dx

+ D!3 L QG L ' P -

2, N

G d Tv

= arctan u

V+ i: , ^&&

dv = ln |v| + C v ln | arctan u| + C ln arctan x + +C x

W 8 "#:*8 '

+ Tdv -

+ -

ǱZ= 4 ǱZ= >$ 7 4 <#:Q#:*8 '

2 N

< u = ln x + i: ^&& =v = − /x < du = dx/x

= dx/x

udv = uv − vdu − dx − ln x − = x x x ln x + x− dx = − x ln x = − − +C x x

-

ln x dx x

=

Tdv = x dx

x arctan xdx =

-

x

x

dx x +

arctan x − - x arctan x − x − + x + x −

=

arctan x −

x

+

x

ǱX: 3 ǱX: ;# 6 3 9! DO!D(7

2 N

< u = arctan x + i: ^&& v = x / < du = dx/(

=

D!3 & {! +a ++a ( %! ^ %+5/< c! ^+.P & q%+ ! TZ ,G ' QG L P 5Q; & SB ;& T5& (& D!3 &/ 4j(& +a ++a & ' P VLH < Ǳ_B & Ǳ_B \< L / %! LO d L D/ = P =V+"& " fX '+8

U& < v = v(x) < u = u(x) + i: =%! B udv ^+5 ,G < P (a; b) & ]KN

+x )

< %!B _+

u dv = uv −

-

= =

du = dt -

t sin t − sin t

Tdv

t sin tdt

−t cos t + sin t + C

=

−(arcsin x) cos(arcsin x) + x + C −x +x+C −(arcsin x)

=

I

=

-

e

x

sin(x) −

(x − )ex + C

# (cos(x))

e

+x

i: &

dt

=

x arctan x −

=

x arctan x −

ln |t| + C

=

x arctan x −

ln | + x | + C

x

10

+x )

udv = uv − vdu dx = x arctan x − x +x -

arctan xdx

2 N

=

-

+ i: TG & x + C < du = − sin(x) dx

=

< dt = xdx V t =

= cos(x) e

xex − ex + C

-

arctan xdx

=

+ Tdv = dx <- u = arctan x + i: ^&& v = dx = x + C _+ < du = dx/(

sin(x) dx - (sin(x)) e x − e x ( cos(x)) dx e x sin(x) − e x cos(x) dx

+ Tdv -= e x < u ^&& =v = e x dx =

=

udv uv − vdu x xe − ex dx

=

x

=

− cos tdt

-

=

x

2# N

e

2

-

+ Tdv =-e x dx < u = sin(x) + i: ^&& v = e x dx = e x + C < du = cos(x) dx =

`b)-!

< u = x V+

i: ^&& v = ex dx = ex + C < du = dx + xe dx

=

=

I

= ex

-

−t cos t −

v du

^+:X L %! : ^&& < udv = d(uv) − vdu V 2 =V+G& ,G < ^

cos tdt

=

24 ^

d(uv) = udv + vdu

+ Tdv = sin tdt < u = t V+ i: S ^&& v = − cos t <

x arcsin x dx −x

vdu

8 < Q@ 2< & B &

+C

2+ N

x arcsin x dx −x

-

-

dx

< x = sin t 24 ^ Tt = arcsin x + i: ^&& =dx = cos tdt -

%

t

<#PG 345 . <#:Q#:*8 ''

−

=

e

−x

Re

W 8 "#:*8 '

-

( + i)x + (i − )

−

× cos(x) + i sin(x) = −

e−x

arcsin xdx,

)

x sin xdx,

)

-

)

x cos(x) dx,

-

)

−x

dx,

-

)

x ln xdx, -

) )

e -

√

x

x

dx,

cos xdx,

(x+ )arctan xdx, -

) ln(ln x) dx, x √ arcsin ( x) √

) dx, −x sin x ) dx, ex ) xex sin xdx, ) )

sin x ln(tan x) dx, x

cos( ) dx, x

-

x dx , x − - − dx, ) x

)

) )

=

-

I+ x cos x dx,

-

P (x) Q(x)

x

)

ln xdx,

) )

sin(ln x) dx,

−x +x

x ln -

dx,

x arctan xdx, cos (ln x) dx,

- ) a + x dx,

) ) ) ) )

arcsin -

I @Q5 & T+

e x ( sin(x) − cos(x)) −

x

( sin(x) − cos(x)) + C

sin x

e x ( sin(x) − cos(x)) + C

= dx

a −x

2% N

+ i: −xdx ^&& =v = x < du =

2* N

√ x dx,

ln(sin x)

a −x - I = a − x dx −x = x a − x − x dx a −x a −x +a dx = x a −x − a −x - dx = x a −x − a − x dx − a a −x x = x a − x − I − a arcsin a ^&& < I + I = x a − x − a arcsin x + a x x a +C I= a −x − arcsin a

-

sin(x) − e x cos(x)

− e x sin(x) dx

I= e

+ Tdv

-

ln x √ dx, ) x ) ln x+ +x dx,

)

I=

x ex dx,

QG @D!3 & s7 ( C Z L . −x xe cos(x) dx ,G ,e C & . ! V+ .C L \< &

dx,

-

x dx dx, (x + )

xe

-

=

tan x ln(cos x) dx,

= x

e x ee dx.

9!NF 234 - 9! DO!D(7 %!

e

^&& =V+!

x dx, ex

-

)

(e )(− sin(x) dx)

−( x + ) sin(x) + C

-

)

=

(x − ) cos(x)

+ D!3 L QG L ' P )

$ x

=

%%

= =

: & U& L +GQG W(& ^

" S < j: fkD/ = P 7.B $ Q(x) < P (x)

=

1>

-

dx

cos(x) dx = Re xe e Re xe( i−)x dx ( i−)x Re x de i −

− − i ( i−)x ( i−)x Re − e dx xe

− ( i−)x ( i−)x Re ( + i) xe − e dx

− ( i−)x ( i−)x Re ( + i) xe − e i − − −x e Re ( + i)x + ( + i) e xi

−x

−x

xi

<#PG 345 . <#:Q#:*8 ''

W 8 "#:*8 '

U& 5 < 7.B $ ' . & 5& ^+$ P

< L 8 & ! U& =. _ !

"& L 7 : A =a = < a, b, A ∈ R d T w[Q k

,: L %! : T d %!& Q@ & cos u =

+ cos(u)

^ L 25: & Q@ %! ^8. =V+ ! =" ! ,:

(Q { & (k − ) @D 7.B $ V+S p X α (Q %&E C < P (x)

2 W

-

Ax + B dx (ax + bx + c)k

L Ta =

2

=

A x−

+

=

-

dx x−

= ln |t| +

+

A (ax + b) + Ax + B = a

-

dt + t

-

^&& Tu = ax

− x+ (x − )(x − )(x − )

A ax + b a (ax + bx + c)k Ab +B + a (ax + bx + c)k

=

-

ax + b (ax + bx +

c)k

,< ,G

du uk

dx =

@;& & T< ,G ax + bx + c =

2 N

x

Ab +B a

+ bx + c V+ i:

-

ds s

=s = x + < t = x − %! " i:

x U& ,G (x − )(x − )(x − ) =+ D!3 V+ V+N |( & 24 & =

+

2 W

Ax + B (ax + bx + c)k

ln |x + | + C

x x = (x − )(x − )(x − ) x − x +

du uk

V+ mK!

B x +

dx = x +

-

@! U& L +GQG & B< 7 \< < TΔ = b − ac < < a = V+ V+N ax + b & Ax + B &

ln |s| + C

= ln |x − | +

=

A A = (ax + b)k a

Ax + B (ax + bx + c)k

x + x − S , =x + = A(x + ) + B(x − ) 24 ^ & =A = V
! U& L +GQG & ^&& TV+ ! u = ax + b +a ++a

L _/- % & ( $' FG/ I ( EE-

=+ D!3 V+N & L+ T%! |( L . 24 @B $ = =%! (x − )(x + ) && x + x − T
d

A (ax + b)k -

ax + bx + c

x + U& ,G x + x −

∈ R

_/- % & ( $' FG/ I ( EE-

$ / && < < ^ ^+:X L +GN & mK! < +H α C < P (x) { T< vX < 7.B " D!3 & QG % T^&& =V+ dx =%! ,G

Ta, b, c, A, B

P ^+$ L +GQG %! : ^&& = &

P (x) (ax + bx + c)k− dx +α ax + bx + c

=

=

(ax + b) Ax + B T w (ax + bx + c)k =Δ = b − ac <

=ax + b -

x−

19

=

a

(ax + b) +

ac − b

dx = (ax + bx + c)k =

=

-

tan u

a

ac − b

V+ i: < B ^&&

ac − b

( + tan u) du

{ a (ac − b )( + tan u)}k k− k− / −k a (ac − b ) ( + tan u)−k du k− k−

a

(ac − b )(−

-

k)/

(cos u)

(k−)

du

<#PG 345 . <#:Q#:*8 ''

W 8 "#:*8 '

! P L C. & d %!& U& mK! V+ _

V+ i: p ^ & = A x−

B C + (x − ) (x − ) D E + + (x − ) (x − )

=

(x + ) (x − )

+

x − x + = A (x − )(x − )(x − ) x−

+

B C + x− x−

^&&

+

= A(x − )(x − ) + B(x − )

x

+C(x − ) (x − ) + D(x − ) (x − ) + E(x − )

+B(x − )(x − ) + C(x − )(x − )

Tx = < x = − Tx = − Tx = Tx = / & < = E T = −B V
%!& {+ & Tx = < x = Tx = i: & + = = C < = −B T = A V
⎧ ⎨

A − B + C − D + E =

A − B + C − D + E = ⇒ ⎩ A − B + C − D + E = ⎧ ⎨ A + C − D = − A + C − D = − ⇒ ⎩ A + C − D = − ^&& =D = − < C = - TA = − + dx & %! && dx (x − ) (x − )

-

- #

=

=

= = =

+

x−

x−

ln |x − | − ln |x − | +

x− (x + ) (x − )

x− A = x+ (x + ) (x − )

ln |x − | + C

U& ,G

2 N

=+

; C 8 & B & = +

B C + x− (x + )

=x − = A(x + )(x − + B(x − ) + C(x + ) + T− = −B V x = < x = Tx = − i: & TB = / + =− = −A − B + C < = C ^&& =A = C − B + = − / < C = /

-

+C

2+ N

+

V+ T|( x +

(x − ) =s = x − < t = x − %! " i: x =+ D!3 U& ,G (x − )(x + ) V+ i: p ^ & = x A = x− (x − )(x + )

x+

D!3

+ ln |x − |

−

x dx = (x − )(x − )(x − ) - − / / dx + + + = x− x− x− dx dx dx = dx + − + x− x− x− =

− − + + x− x− (x − ) $ − + + dx (x − ) (x − ) dx dx dx − − + x− x− (x − ) dx dx − + (x − ) (x − ) dt ds dt ds ds − − − + + t s t s s s− − ln |t| + + ln |s| + + +C t s − − ln |x − | +

x + = A(x − )(x − )

−

Bx + C x +

& =x = A(x + ) + (Bx + C)(x − ) + = A V
x− dx = (x + ) (x − ) - − /

/ /

dx + + = x+ x− (x + ) dx dx dx = − + +

x+ (x + ) x − dt dt ds + = − +

t t s − + ln |s| + C = − ln |t| +

t

= −

ln |x + | −

(x +

)

+

ln |x − | + C

=s = x − < t = x + %! " i:

D!3 U& ,G (x − ) (x − ) =V+

2 N

16

<#PG 345 . <#:Q#:*8 ''

W 8 "#:*8 '

^&& Tx = tan u V+

-

u

=

+

sin(u)

u

+

I

+

-

+ tan u x = + x /

cos (u) =

− sin (u) −

=

D!3

=

ln |x − | −

ln |x − | −

=%! " ! t = x

(x

=x

x(x + )(x + )

U& ,G

dx x(x + )(x + )

√

Ci + D

x +

=

z(z + )(z + )

=

A B + z z+

2* N

+ +a ++a L T< ,G

U& ,G

2# N

=

−

i<x

^&&

i: & < (Ai + B)(− + ) = − +

=

= i

−

-

I

xdx x (x + )(x + ) dx z(z + )(z + )

+

TB = + = Ci + D = − < Ai + B = T^&& ^&& =C = < D = − TA = x dx (x + )(x + ) dx dx − = x + x + √ V < ,G x = tan u i:

=

I

= = =

C D + z + (z + )

=

+

A(z + )(z + ) + Bz(z + )

arctan x −

( + tan u) du tan u +

du √ u + C arctan x − √ arctan(√x) + C arctan x −

arctan x −

2% N

=+ D!3 U& ,G (x + ) i: %! : ^&& =%+ '+8 & L+ =

+Cz(z + )(z + ) + Dz(z + )

1F

&

√

√ -

C & T%! U& ' " _+ 4 ,G V+ i: mK! T; =

-

√

-

=

dx +

= (Ax + B)(x + ) + (Cx + D)(x + ) √

=+

x |( < 24 & p ^ & = + =z = x V+ i: mK! < V+ -

dx x + x +C ln |x + | + arctan x +

V
(x − ) (x + )

arctan x + x x(x − ) + +C (x + )

(x) −

x Cx + D Ax + B + = (x + )(x + ) x + x +

+x =

=+

V+ i: p ^ & =

x

x + )

x

x (x + )(x + )

D!3

+C

^&& =

I =

tan u

=

dx (x + )

xdx (x − )(x + ) dx x− − dx x + x +

^&& Tx − =

sin(u)

= =

V Ttan u = x/ 8 & B &

sin(u) =

-

-

( + tan u) du dx = (x + ) ( tan u + ) du = = (cos u) du ( + tan u) - + cos(u) = du = ( + cos(u) + cos (u)) du = du + cos(u) du + cos (u) du + cos(u) u sin(u) = + + du

+

<#PG 345 . <#:Q#:*8 ''

= x −+ −

−

x +

(x + )

+

W 8 "#:*8 '

(x + )

+

%!& z = < z = − Tz = − Tz = i: & = A + < = D T = −B T = A V
(x + )

(x + )

& =V+G& ,G 74 P L %! LO dx V+ i: p ^ & T24 ^ In = (x + )n V TǱ_B & Ǳ_B \< L ! -

In

=

(x + )n

= = =

-

-

x+

< In+ = n − n

In +

n(x

x + )n

-

-

x +

2, N

x + x +

=+ D!3

U& ,G 8 & B & =

= x + x +

− x √ = (x − ) − ( x) √ √ = (x − x + )(x + x + )

^&&

V+

dx = arctan x + C x + dx x = = I + (x + ) (x + ) x = arctan x + +C (x + ) dx x = = I + (x + ) (x + ) arctan x + x x = +C + (x + ) (x + ) dx x = = I + (x + ) (x + ) x x = arctan x + + (x + ) (x + ) x + +C (x + ) dx x = = I + (x + ) (x + ) arctan x + x x = + (x + )

(x + ) x x + + +C (x + ) (x + )

I

dx = x(x + )(x + ) dz dz − + = z z+

dz dz + z + (z + ) − = ln |z| − ln |z + | + ln |z + | + +C z+ +C = ln |x| − ln |x + | + ln |x + | − x +

nx dx (x + )n+ x (x + − ) dx + n n (x + ) (x + )n+ n dx n dx x + − (x + )n (x + )n (x + )n+ x + nIn − nIn+ (x + )n

=

=

I

I

I

I

-

dx (x + )n

-

x + x +

-

+

− x +

x

(x +

)

√

x + x +

x − √

√

x +

−

√

x

√

x − x +

√

V s = x < x − = t i: & 24 ^ √ √ - xdx x + x + x + dx = dx −

x + = x− -

= =

x +

√

dx √ − +

√

dt

t + −

=

√ -

√

√ -

x + xdx (x ) +

ds

s +

arctan t − arctan s + C √ √ arctan x − − arctan(x √

)+C

2_ N

=+ D!3 x U& ,G (x + ) V+P s& x + { & 24 & =

x x dx = − x + I − I (x + ) + I − I + I x

= =

^&&

=

x +

x ((x + ) − ) = (x + ) (x + ) (x + ) − (x + ) = (x + )

arctan(x) − x (x + ) x x − + +C (x + ) (x + )

+ +

1I

(x (x

+ ) − (x + ) + ) − (x + ) +

<#PG 345 . <#:Q#:*8 ''

W 8 "#:*8 '

V+ T(x x + (x − x + )

− x + ) = x −

2 N

8 & B & <

,G T0=6=6 L w0x \< '. & dx =+ D!3 (x + x + ) V+ i: p ^ & =

x− x − + (x − x + ) x −x+ x− x − − + (x − x + ) x −x+ −

= =

−

x −x+

-

="& ( −

^&&

-

x + dx = x − I + ln |x − x + | − I (x − x + ) x− dx d

=I = dx < I = (x − x + ) x −x+ √

√

(x −

=

P (x) x− = +α (x − x + ) x −x+

-

+α(x + x + )

< ! < x && { / & T^&& V TO& ⎧ A − B + α = ⎪ ⎪ ⎪ ⎨ A + B − C + α =

dx x −x+

=E

V+ i: V+ mH T%! (Q E $

D = / TC = − / TB = − / TA = − / ^&& + TN ^ < +Q< i: & B & =α = / < V+S

= (Ax + B)(x − x + )

< ! < x && { / & T^&& V TO&

=B

−A − B + α = A − C − α = B + C + α = −

⎧ B+ ⎪ ⎨A = − ⇒ C = − B+ ⎪ ⎩ B− α=

V+ i: V+ mH T%! (Q B $

=α = −/ TC = − / TB = TA = −/ ^&& V I +Q< i: & B & T^&&

=

− x − dx − x −x+ x −x+ x + − x − x + − I

-

(x +

x + dx (x − x + )

)

+C

,G

2 N

& 24 T%+ |( @B L . 24 @B $ = V+ V+N d |( x + = (x − x + )

x +

= (x − x + ) x +

= x + ln x − x + + x − x + √ √ arctan (x − ) + C −

arctan

=+ D!3

V . <

-

− x − x − x + x dx = (x + x + ) (x + x + ) dx + x +x+ − x − x − x + x = (x + x + ) √ √ +

=

=

-

=

I

⎧ A = − / + E ⎪ ⎪ ⎪ ⎨ B = − / + E ⇒ C = − / + E ⎪ ⎪ ⎪ ⎩ D = / + E α = /

B − D + α = ⎪ ⎪ ⎪ ⎩ C − D − E + α = −E + D + α =

−(x − )(Ax + Bx + C) + α(x − x + )

#

(Ax + Bx + Cx + D)(x + x + )

+C

="& ( − ) = @D 7.B $ ' P (x)

& =P (x) = Ax + Bx + C V+ i: T^&& V+S + T " i: @;& ^+:X L +GN x−

@D 7.B $ ' P (x)

V+ i: T^&&

−(x + )(Ax + Bx + Cx + Dx + E)

T^&& =V+ ! 0=6=6 w0x \< L & I < V+ i: -

dx x +x+

V T " i: @;& ^+:X L +GN &

)

) =

-

P (x) = Ax + Bx + Cx + Dx + E.

S P %Q! &

I = arctan

P (x) dx = +α (x + x + ) (x + x + )

x − x

x − x

+ x − (x − x + )

+ x − T ! P & +! Kp & TmK!

V+ V+N x

x − x

+

− x + &

+ x − =

= (x − )(x − x + ) + − x.

11

<#PG 345 <#4 A1 #$ #*1 >$ 7 ,'

) )

-

x + x +

dx,

dx , (x + x + )

) )

-

W 8 "#:*8 '

+ D!3 L QG L ' P

dx , x(x + )

-

x dx . (x + x + )

234 9!3 >0!# !(0 ;# 6

*% 9!NF

)

V+G& ,G P (x)/Q(x) U& L V+P(& + i: V+ i: ="& P (x) @B L S_& Q(x) @B d Q & < @B < ' @B C & Q(x) @_ ="& Q(x) = (R (x))n (R (x))n · · · (Rm (x))n 8" & U& +GQG 8!S< ! \< '. & D G U& ' L +GQG & P (x)/Q(x) ^"< ="& R (x)R (x) · · · Rm (x) d |( . ="& ! +& U& ^+$ L +GQG %! =%! ! L @+A/ & \< ^ ,4

)

)

)

P (x) dx Q (x)

-

)

)

!

dx

-

)

)

− x − x x − x + x − x +

-

-

)

(x) = R (x)R (x) · · · Rm (x)

2

dx x + x + x + x +

-

)

) Q (x) = (R (x))n − (R (x))n − · · · (Rm (x))nm −

+ D!3 L ,G

-

)

-

)

,

(x + )(x + ) dx, (x + )(x + )

-

deg(P (x)) < deg(Q (x))

-

)

7.B $ T G d = De < Y+34 C P ni T"& B< X Q (x) < Q (x) TP (x) TP (x)

)

(x + )(x + x + )

)

Q(x) = (R (x))n (R (x))n · · · (Rm (x))nm

deg(P (x)) < deg(Q (x))

-

)

@B C L &A74 24 & Q(x) @_ < P 8" & Q & < @B < '

)

-

)

deg(P (x)) < deg(Q(x))

) Q

-

)

& 7.B $ Q(x) < P (x) + i:

-

-

)

( $' FG/ (G 2Q&@ !

P (x) P (x) dx = + Q(x) Q (x)

-

)

m

-

dx , (x − )(x + )(x + ) x + x −

dx, (x − )(x + )(x − ) x −x+ dx, (x − x + ) x + x +

dx, (x − ) (x + ) dx , (x − x + )(x + x + )

dx

)

)

,

xdx , (x + )(x + ) x + x

dx,

x − x +

dx , (x + )(x + )

)

x − dx, x − x

)

x + x + dx, x(x + )

)

dx

x +

)

,

dx , (x + ) xdx x +x +

) ,

)

x + dx, x(x − )

)

x − x + x +

)

dx,

dx , x(x + )

)

dx , x (x + )

)

dx , (x − )

)

-

x − x +

dx, x − x + dx , x(x + )

x

x − dx

, x + x dx

, x + x − dx , x(x + )

x + dx, x(x − ) x − x +

dx,

x dx , (x − ) dx , x + x x dx , (x + x + )

1J

dx,

Strogradsky½

<#PG 345 <#4 A1 #$ #*1 >$ 7 ,'

W 8 "#:*8 '

= -

/

TC

=

TB

=

/ TA = / + + =E = /

8" & " U& |( 8 & p =

x + dx = (x + )(x + ) x+ x+ = + dx (x + )(x + ) (x + )

x − x + x − x +

i: T8!S< ! @+A/ & B & _+ < . _ V+

'+8 \< '. & T%! %.! ,G @D!3 & V+

x+ + =− (x + ) (x + )(x + ) -

-

ln(x

=+ D!3

-

x + (x + ) V . TmH

− x − x

arctan(x) + C x − x + dx ,G

2 N

-

=

=

x −x+

-

O& < ! < x && { =V+! V+S + < / &&

-

⎧ ⎪ ⎨A = B= ⇒ ⎪ ⎩C = D=

-

+

dx + x −x+ x −x+ √ √

x +

arctan

(x −

x + dx ,G (x + )(x + )

)

+C

2 N

@+A/ < " U& |( @_ & B & = V+ i: T8!S< !

+x (x − x + )(Ex + F x + G)

−x − x − + x (x − x + )

A(x − x + ) − (Ax + B)(x − )

dx =

x +

=+ D!3

−(x − x + x)(Ax + Bx + Cx + D)

x − x + dx = x (x − x + )

− x − x

(x − x + )

x(x − x + ) = (Ax + Bx + C)x (x − x + )

-

Cx + D dx x −x+

+

< & T^ :S } |( < ^+:X L +GN L mH

⎧ A=− ⎪ ⎪ ⎪ ⎪ B = − ⎪ ⎪ ⎪ ⎨C = ⇒ D=− ⎪ ⎪ ⎪ E= ⎪ ⎪ ⎪ ⎪ ⎩F = − G = −

=

⎧ C= ⎪ ⎨ −A + D − C = − ⎪ ⎩ −B − D + C = − A+B+D =

Ax + Bx + Cx + D x (x − x + ) Ex + F x + G + dx x(x − x + )

⎧ −D = ⎪ ⎪ ⎪ ⎪ F −E−A= ⎪ ⎪ ⎪ ⎨E = A + B − C + F − G = ⎪ ⎪ ⎪ D − C = − ⎪ ⎪ ⎪ ⎪ ⎩ C − D + G = −B + E − F + G =

+

O& < ! < x && { =V+! V+S + < / &&

x − x + dx = x (x − x + ) =

Ax + B x −x+

+(Cx + D)(x − x + )

+ )+

x (x − x + )

dx =

< & T^ :S } |( < ^+:X L +GN L mH

@+A/ < " U& |( @_ & B & = V+ i: T8!S< ! -

− x − x

(x − x + )

x + dx = (x + )(x − x + ) ln |x + | x+ − = (x + ) +

= (x − x + )

x + dx = (x + )(x + ) Ax + B Cx + Dx + E = + dx x + (x + )(x + )

< & T^ :S } |( < ^+:X L +GN L mH x +

= (x + )(−Ax − Bx + A) +(x + )(Cx + Dx + E)

O& < ! < x && { =V+! V+S + < / &&

+

⎧ C= ⎪ ⎪ ⎪ ⎨D −A = E + C − A − B = ⎪ ⎪ ⎪ ⎩ D − B + A = E + A =

−x − dx x(x − x + )

J)

?$ =7 [ . <7\= R 345 . <#:Q#:*8 2'

√ -

=

(x + ) + dx =

√ -

=

√ - (t) + dx

'+8 \< '. & T%! %.! ,G @D!3 & V+

+ t dt

−x − x − =− + x x(x − x + ) x −x+

ln t + +t +C

√ t

=

W 8 "#:*8 '

+t + √ x+ x+ = + √ x+ x + +C + ln + + x+ = x + x + √ √ + ln (x + ) + x + x + + C

V Tu + -

=t

mK! < u = x i: & -

V . TmH -

− ln |x| + ln(x − x + ) x (x − x + ) √ √ − arctan (x − ) + C

= −

2 N

\< & L QG L ' P ! + D!3 8!S< ! x dx dx ) ) ,

du/ x dx = x + x + u + u + du = (u + ) + dt = = ln t + t + + C t + ln x + + x + x + + C =

)

-

x

+ x + dx,

- ) x + x +

)

dx, )

dx

x

+ x +

x

+

-

-

cos x dx + sin x − cos x dx

-

(x + ) (x + )

dx , (x + ) (x + )

x + x + x + dx, x (x + x + )

1%

? ) "

dx ax + bx + c

V+ .C − ' x { T,8 L |a| L +S : & w =V+ =V+ U& ,8 L 4 2DC w0 ,8 L 2DC T = b − ac < a %fC & & w> < (x + α) − β T(x + α) + β 24 ! L 8 & ,G Tx + α = βt i: & =" P β − (x + α) =0=6 ,<B fkD/ S D L : W" L 8 & " :5 - x +

-

)

dx. x (x + )

-

7 8" & QG @D!3 & - L 24 & ax + bx + c dx <

x + dx,

,

8" & ,G @D!3 & 7.B $ ' Pn (x) d

x −

<# :6 + Z

, x ln x + ln x + ) ex + ex − e x dx, )

,

,

dx ) x x + x − dx, ) , x + x − dx ) , ) dx . x − x − x + x

) sin x cos x + sin x − dx, -

x (x + )

- 96Y: +P 234 - 9! DO!D(7

-

)

-

)

"

-

-

)

+ D!3 L QG L ' P

x − x + dx = x (x − x + ) x + x +

? ) "

-

Pn (x) dx ax + bx + c V+ i: =%! n @B

dx x +

- x −

dx, ,

V Tx +

-

Pn (x) dx = Qn− (x) ax + bx + c ax + bx + c

x

J

dx,

-

+ x +

dx dx, x − = t i: &

dx

2

=

-

− x dx, dx −x

.

"

√ - x + x + dx

?$ =7 [ . <7\= R 345 . <#:Q#:*8 2'

W 8 "#:*8 '

-

8 < x + x + =

A(x

+λ

{ & (n − ) @B 7.B $ ' Qn− (x)

^+:X L mK! =%! , C _+ λ < "& , { T< ^+:X / 3 & < V+S M %.! ,G ! =V
+x+ )

+(Ax + B)(x + ) + λ

vX < < { / && & = A + < T = A + B T = A V TO& < =λ = / < B = / TA = / T+ =B + λ ^&& -

x + x + dx x +x+

=

+

V x + -

-

=+ D!3

x +x+

-

dx

x +x+

t i: &T %! %.! ,G

=

= √

dx √ (x + ) + ( )

)

) ) )

-

x dx x −x+

dx

x dx dx x + x− x + dx x + x

x −x+ dx − x − x

) ) ) )

-

dx −x

−x

x

−x

=

(Ax + Bx + Cx + D)( − x ) −x(Ax + Bx + Cx + Dx + E) + λ

vX < < { / && & TA − C = T−B = T−A = V TO& < + =D + λ = < C − E = TB − D = < E = −/ TD = TC = −/ TB = TA = − / ^&& =λ = -

x dx x −x+

x dx =− −x

(x + x + ) −x +C -

x + x + x +x+

x + x + dx x +x+

=

2 N

,G V+ i: =

(Ax + B) x + x + +λ

x + x −

dx x +x+

V TO& < ^+:X L +GN & x + x + =A x +x+ x +x+ x + +(Ax + B) x +x+

x dx x + x + − x + x

−x

8 <

-

−x

λ

+

! "

x + x

V+ i: =

+(Ax + Bx + Cx + Dx + E)

x + x + = (x + ) x + x + x +x+ + ln x + + x + x + + C

-

"

= (Ax + Bx + Cx + D) −x

x

dx

+ D!3 L QG L ' P

2

= (Ax +Bx +Cx +Dx+E) −x

=+ D!3

-

,G

V TO& < ^+:X L +S M &

V . TmH -

x dx −x

+λ

√ √ ( t) + ( ) dx = = ln t + t + + C t + √ = ln + + C x+ + x+ = ln x + + x + x + + C =

x dx −x

-

√

dx x +x+ -

x+

dx ax + bx + c

dx

J0

λ + x +x+

?$ =7 [ . <7\= R 345 . <#:Q#:*8 2'

W 8 "#:*8 '

8" & ,G @D!3 & 2DC |( < 24

+ (x + ) − x = (Ax + B) −x −x λ +(Ax + Bx + C) + −x −x

8" & 4 TV+ ax + bx + c ,G =& P 9=I=6 n< ,G

< A − C = T−B = − T−A = − + < C = − / TB = / TA = − / ^&& =B + λ = + =λ = / -

= −

)

) )

x − x + −x + x − x + +

-

(x + ) x +

dx,

(x − ) x +

dx,

)

x

-

-

dx,

=

-

dx (x + ) x +

= =

dx.

< B +D = T A+ C = TB = TA = + D = TC = / TB = TA = / ^&& =C + λ = + =λ = − <

dx (αx + β)n ax + bx + c

-

2

t

−tdt

=

− t +

"

−dt/t ( /t) ( /t − ) + −t dt t − t +

=+ D!3 -

-

t

− t +

A(t − )

t

− x dx ,G (x + )

− x dx = (x + )

dx

t

− t +

+

t

2 N

V 24 ^ =

=

-

(x + )( − x ) dx −x −x − x + x + dx −x

V+ i: TmH

− t +

-

+ =

-

A t − t + +λ

−t

x + dx = x dx x + x +− = x + x = (x + ) x + − ln x + x + + C x

V+ i: -

(Ax + Bx + C)(x + )

+x(Ax + Bx + Cx + D) + λ

? ) # "

&

(Ax + Bx + Cx + D) x + dx +λ x +

x + = (Ax + Bx + C) x + x λ +(Ax + Bx + Cx+ D) + x + x +

x

" n; ,G +D" QG 4 =V+ ! =& P 9=I=6%./ = /t i:

=+ D!3 V 24 ^ =

+

(x + ) x + dx,

(x + x + ) x + x −

V x +

x + dx =

"

(x + ) x −

"

V+ i: TmH x

-

2

,G

- x + x x + dx = dx x x +

−x

x + x + dx,

(x + )

x + dx

dx

arcsin x + C

)

8" & ,G @D!3 &αx + β = /t +a ++a L

-

-

+ D!3 L QG L ' P

)

− x dx = (x + ) = −

? ) " "

P (x) ax + bx + c dx

λ − t +

J>

− x dx (x + )

=

(Ax + Bx + C) −x dx + λ −x

AR 4 345 . <#:Q#:*8 I'

W 8 "#:*8 '

D!3 L QG L ' P -

dx

-

)

-

)

+P

=

−

(x + )dx (x − x + ) x −

.

=

−

x −

)

,

dx (x − ) x + x +

,

)

-

dx

x + x

(x + )

, -

, (x + ) −x (x − )dx ) , x + x + x

2

+ i: 24 ^

+

a

2R

x,

-

H%

pi qi

+

t

. mH

+

(x + )

= −

x+

/t i:

+C

&

u/

−

=

< ad = bc

−

−dt/t

−t

−t

V+ i:

+ =

u/ x −

x + x x

+ u/ + C / − − /

x

2 N

( /t) ( /t) − t dt

(t ) tdt dx =− x x − −t ( − u) ( − du) √ = − u u−/ (u − ) du = = u−/ (u − u + )du - u/ − u/ + u−/ du = =

J9

−

+

−

-

} A ^ 8$ n TV+ ! ax + b = z n cx + d =%! = = = < q Tq

dx (t − ) +

− t + + C

dx x x −

? )

p /q p /q ax + b ax + b , , · · · dx cx + d cx + d

+a ++a L T P Y+34 C L P

ln t −

24 ^ Tu =

2

− t +

− t +

=

= maVm + (m − )bVm− + (m − )cVm−

t

√

V x =

ax + bx + c =

8" & QG D!3 &

t

+ +C x+ (x + ) √ −x+ x + x + − ln (x + ) x+ -

(b − ac)V

+>P 3 234 - 9! DO!D(7

−

dx

√ -

− t +

dx = (x + ) x + √ = − ln − x +

=

7.B $ < αm %&E C T m P L & Vm = B< X Pm− (x) (m − ) @D =Pm− (x) ax + bx + c + αm V m P L & xm−

t

−

b V V = ax + bx + c − a a V = (ax − b) ax + bx + c a

x

−

dx

−

=

[Qx

P

= − t + − t − t +

t

dx , (x + ) x x − m =Vm := x dx ax + bx + c

-

−tdx

,

(x − )

-

)

(x − )(x + ) x +

dx

-

)

-

dx , (x + ) x + x +

-

)

+

dx , (x + ) x + x +

)

)

=−A + λ = < A = − ^&& =−t = A(t − ) + λ < A = λ = − / mH

"

/ x

+C

+ x −

+C

)P8#0 <)= $ . <#:Q#:*8 J'

W 8 "#:*8 '

2

&& ax + b TL ,G cx + d i: < n = ^&& =q = < q = T%! x − + =x − = z V+

m+

V+ i: G d T"& Y+34 C S w| n ,G T4 =%! p |( k d Ta + bxn = tk =& P U& '

- √ x − dx √ x −

m+

+ p S w V+ i: G d T"& Y+34 C n T4 =%! p |( k d Ta + bxn = xn tk =& P U& ' ,G

2

=q

-

x/ − x/ + x/ + C = Y+34 C ' p = − L ,G } |( $ mH =%! d W+H wx 5 =%! V+ i: T%! k = && m = − < n = / ^&& < dx = t dt < x = t − + T + x = t

2 N

− dx = x− + x/ dx √ x( + x) dt − (t + ) .t dt = = t t(t + ) - − = − t t+ (t + ) =

ln |t| − ln |t +

=

√ ln |x| − ln( x + ) + √ x+

T%+ Y+34 p

|+

t+

z z dz = z

2 N

ax + b +x = L ,G cx + d −x +x = = z V+ i: < n = ^&& −x zdz − <x= z + < dx = (z + ) z +

<

zdz + x dx (+z ) = z −x −x − zz − + - = − dz = z − arctan z + C z + ⎞ ⎛ +x + x ⎠+C − arctan ⎝ = −x −x

)

+C

-

√ √ x+ x √ √ dx, x − x x

+C

)

2 N

)

)

z dz

-

)

-

z + C = (x − )/ + C

+ D!3 L QG L ' P

L ,G − / + m+ = w|x %Q T5 =%! Y+34 = n / k = && p |( 8 & B & mH =%! d W+H √ < x = (t − ) + = + x = t V+ i: T%! ^&& < dx = t (t − ) =

=

^&&

√ x − x dx = x/ − x / dx = x/ − x / + x/ dx

-

= =

C ' p = L ,G ^&& =%! d W+H w[Qx %Q 5 =%! %De Y+34 -

-

-

+

dx

+ x

) ,

)

dx

, ) (x − ) (x + ) dx , ) (x + ) (x − ) x dx √ , x−

)

-

-

x+ x−

dx (x − )

−x +x

dx

x − √

x+

(x + ) −

√

√ / + x √ + x/ dx dx = x−/ x = (t − )− (t )/ t (t − ) dt = (t − )t dt = t − t + C

,

x − + √ dx.

−

x+

' N7!/ 9 ': # - 9! DO!D(7

-

,

dx , ( − x) −x √

-

dx,

I%

< P S C p < n Tm V+ i: - ? ) 8 I ,G T24 ^ =I = xm (a+bxn )p dx %! D!3 &/ L %Q $ L n & a+bx G d T"& %De Y+34 C p S w[Q =V+P ,.5 M&; < + ! p V+ i: G d T"& Y+34 C p S w T4 =%! n < m < } |( k d Tx = tk =& P U& ' ,G

+ √x/ − ( + √x)/ + C = m+ = − < p = − L ,G n m+ + p = − T + Y+34 C =%! Y+34

2 N

n

J6

W 8 "#:*8 '

#Q$ <L#X* #X5 V'

|( 8 & B & mH =%! d W+H wx %Q T5 + = + x = x t V+ i: T%! 0 && p ^&& < dx = − t (t − )−/ dt < x = (t − )−/

V+ .C L \< ! L 8 & V+ i: G d T < a S w √ ax + bx + c = ±x a ± t

-

V+ i: G d T < c S w0

x

√ ax + bx + c = ±xt ± c

G d T"& ax

=

+ bx + c = @Q5 @ ' Tα S w> . ax + bx + c = (x − α)t V+ i:

2

-

=

+ -

− + (t + )

dt

+C x + + x + x +

2 N

dx p f ,G x x +x+ V+ i: V+ ^&& c = > S Tx + x + = (xt + ) + x + x + = tx + =x(t − ) = − t ^&& < x + = xt + t =dx = t − t + dt < x = −t + + (t − )

^&& =t = -

dx

-

x

t − ( x +x+

− )

dt (tt −t+ −)

= − t − t + x x +x+ t t − t − dt = ln |t − | + C = t − = ln x + x + − − x − ln |x| + C -

−/

(t − ) dt = −

√ √ x( − x) dx,

=%+ &/

−t

t

+

+ x = x

t

(t − )−/ dt

−

t

+C

+ x x

-

)

- √ sin x dx +P

!O# 9M! V( ! V4

2 N

dx ,G ( x − − x ) T%! x − − x @Q5 @ ' x = S + = x − − x = (x − )t V+ i: ^&&

p f

t t −

dx √ √ , x ( + x ) √ √ - - + x + x √ √ dx, ) dx, ) x x dx ) x + x dx, ) , x +x √ dx ) x + x dx, ) , x + x dx x dx

) dx, ) , ( + x ) + x dx dx √ ) , ) , x +x x +x dx dx ) √ ) , , √ x ( + x )/ x + x x dx dx ) , ) , +x + x - ) x + dx, ) x − x dx, ( +x ) - dx

) , ) x − x dx. x + x - √ ) tan x dx )

+C ln |t + | + t+ ln x + + x + x +

=

(t − )/

+ D!3 L QG L ' P

t + t+ dt (t+) t − + t − (t+)

= + x + x + - t + t + = dt = t+ (t + )(t + )

x− ( + x )−/

=t =

< x + = t − xt Tx + x + = (t − x) + + =dx = t + t + dt < x = t − ^&& (t + ) (t + ) -

+ x

x + x + = t − x

dx

-

=

= −

p f L ,G # V+ i: mH Ta = > S

-

dx

8" & QG D!3 & ? T%! S 5& y = P (x) P x, ax JF

2

U%

) #

+ bx + c dx

]%PG $ ]%1 <@85 . <#:Q#:*8 '

-

um ( − u )k du =

= -

sin

k+

W 8 "#:*8 '

^&& = − x = (x − )t T(x − )( − x) = (x − ) t + =dx = −tdt < x = t +

x cosn x dx

(t + )

(sin x)k cosn x sin x dx

=

-

( − cos x)k cosn x d(cos x) − ( − u )k un du

= =

=V+G& ,G 4 U& L %! :
2

L ,G + < u = cos x V+ i: ^&& =%! m = =

u

=

−

u

+C =

cos x −

-

=

=%! m

=

−

u

sin x −

+

u

-

cos x + C )

2 N

)

− =

)

2 N

(u + u − u (u − )

− ln |u − | u− ln |u + + + u+ ) u + + ln +C u −

)

−x d

x−

#

+

-

dx , x − x + x +

-

dx −x −

,

- ) x + x − dx,

)

-

)

x + x + ) , x + x + x + x+ x +x+ ) +x+ x +x+

< n = − L ,G + < u = sin x V+ i: ^&&

(u −

dx

-

sin x sin x dx = cos x dx cos x cos x u du sin x cos x dx = du = (cos x) ( − u ) u du = − (u − ) (u + ) - = − + + (u − ) (u − ) u− − + − du (u + ) (u + ) u+ =

-

dx , ) (x + ) +x−x

, (x − x ) (x + x + ) ) dx, x + ) dx , x+x -

) x x − x + dx,

+C

sin x + sin x + C

=

D!3 L QG L ' P

sin x cos x dx = sin x(cos x) cos x dx = u ( − u ) du = u (u − u + )du u

x − − x x−

=t =

-

=

^&& =%! m = < n = L ,G + < u = sin x V+ i:

-

dx dx = ((x − )t) ( x − − x ) −t dx t + (t +) = = − dt t {( tt ++ − )t} = − − + t + C

t x− −x − +C =

−x x−

-

sin x cos x dx = sin x cos x sin x dx = ( − u )u (−du) = u (u − )du

t +

-

-

−x+x +x−x

dx,

dx , ( + x + x)

x−

dx.

9=74 - 9! DO!D(7

%

[$ NF # [$ 0 _ %Q $

-

sinm x cosn x dx L +GQG Q@

=V+P / !& S 4W G &> 2Q 7[ n m G I @ V+ i: T"& : m S < u = sin x V+ i: T"& : n + < u = cos x

+

-

|+C

m

k+

-

sin x cos x dx = sinm x(cos x)k cos x dx = sinm x( − sin x)k d(sin x)

JI

]%PG $ ]%1 <@85 . <#:Q#:*8 '

W 8 "#:*8 ' 2B5

` I NA8 &4+ ^

m+n

=V+ ! u = tan x +a ++a L %Q ^

2

< n = − Tm = %Q ^ -

sin x dx = cos x u

=

-

+C =

sin x cos x

dx = cos x

cos x sin x

=

-

sin x cos x

×

2 N dx cos x

=

-

-

< n = − Tm = − %Q ^

45W G 2B5 -

2 N

+ <

-

cos x sin x dx = =

=%! tan x =

,: <

-

=

,: < u = tan x +a ++a L ,< +a ++a L < < V+ !

=

tann x dx

−

=V+ ! cot V tan x U& -

"

p ,G

cotm x dx

cos x u = cot x

-

m = −n @

-

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x=

sin x

J1

-

x−

m

@

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cos(x)dx −

sin (x)

sin(x) − sin (x) + C ^&& < m = < n = %Q ^

+C

2 N

- − cos(x) dx sin x dx = (sin x) dx = - cos (x) − cos(x) + dx = + cos(x) = − cos(x) + dx =

tan x − tan x + x + C

=

-

=

n

dx - sin (x) − sin (x) cos(x) dx − cos(x) dx − sin (x) d(sin(x)) x−

−

tan x dx = tan x dx − cos x dx − tan x dx = tan x cos x - = u du − dx − cos x u −u+x+C = u du − du + dx =

sin(x)

=

& 6

2

- 45W G NA8

` I

D/ & T<_Q 24 < . ! L Q: L V+ B + cos(x) − cos(x) cos x = , sin x = , sin(x) . sin x cos x = < m = < n = %Q ^ ^&&

√ √ tan x tan x dx = dx sin x cos x sin x × cos x cos x √ √ du √ = u + C = tan x + C = u

%Q ^

−

+

(u − ) (u + ) (u + ) + + + u (u − ) (u + ) u + ln |u − | − |u + | + C u − u − u − ln + +C u + u (u − )

=

^&&

tan x + tan x − cot x + C

m + n = −

−

u + u + ( + u ) du = du = u u u + u − + C = u ++ du = u u =

sin xdx

=

sin x cos x −d(cos x) du = =− ( − cos x) cos x u (u − ) - # = − + + u u (u − ) $

u du

×

cos x

-

dx -

-

2 N

< n = − L ,G + < u = cos x V+ i: ^&&

sin x cos x

tan x + C

= −

-

^&& < m + n = −

-

dx

4W G

!

< m + n = − < n = − Tm = − %Q ^ -

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dx √ , tan x

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sin x √ dx. cos x

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P sin x, cos x dx x

sin x =

u +

,

−u cos x = , u +

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dx =

du u +

-

-

= = = =

, = =

P (− sin x, cos x) = P (sin x, cos x)

− ln | cos x| + C

tan x − ln | cos x| + C

-

2 N

dx − -sin x dx − cot x dx cot x sin x u (−du) − du − sin x dx + dx − u du − - sin x − u du − udu + dx

= −

cot x

u

+

u

+x+C

cot x +

cot x + x + C

D!3 L P ,G L ' P

2 N

-

dx

)

u u + u − u +

,G S

u

tan x −

= −

-

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)

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)

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? )

)

n" & N @;& JJ

) )

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sin x cos x dx,

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−

cot x dx =

=

sin x du dx = − sin x u + udu (u + ) (u + ) - + − du (u − ) u + − − arctan u + C u− − −x+C tan(x/) −

-

=

du dx u + = u −u + sin x + cos x + u + + u + du = = ln |u + | + C u+ x = ln tan + +C

V = 0=6 '. &

-

u

.C L 24 & cot x U&

]GB & 24 ^ =u = tan V+ i: V+! U& ' L QG & L u

tan x tan x dx − tan x dx cos x dx − tan x dx tan x cos x u du − tan x tan x dx u du − tan x dx − cos x u du − udu + tan x dx

-

cot (x)dx,

-

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sin (m − n)x + sin (m + n)x cos (m − n)x − cos (m + n)x sin(mx) sin(nx) = cos(mx) cos(nx) = cos (m − n)x + cos (m + n)x

-

-

-

sin( x) cos(x)dx = = sin(x) dx +

−

=

cos(x) −

-

sin(

cos(

sin(x) + sin(

sin( -

= =

x) + C

2 N

(sin(−x) − cos(x)) dx sin(x) dx − cos(x) dx

cos(x) − sin(x) + C V ,: ^+! '. &

-

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−

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sin x +

) )

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dx , sin x + cos x

)

sin(x) +

=

D!3 L QG L ' P

-

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=

√ -

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x) dx

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x)

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=

sin x + du = u + √

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dx dx = (x + ) (x + ) + (x + ) x + x + dt ( + tan t) dt cos t = = sin t (tan t) (tan t) + cos t cos t d(sin t) − cos tdt +C = = = sin t sin t sin t − − x + x + = +C = +C √ tan t x+

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V x +

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(x + x − x + ) sin(x) dx

+(−A x + (B − A )x + B − A ) sin(x)

+ B = (A + B ) = A + B = −A = (B − A ) = B − A = TB = / TA = " + G ! ^ L

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(x + ) cos(x − ) dx (x − ) sin x cos(x) dx

(" S 4 ǱX: 3 ǱX: !"

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v (x) = vn (x) =

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V T< ^ ^+:X L +GN &

v (x)dx,

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vn− (x)dx

+(B x + B x + B ) cos x

24 ^ -

+(B x + B x + B x + B ) sin x + C

V+ [5 <

·········

(x + x + ) cos(x) dx =

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=

x sin x cos(x) dx

-

V T< ^ ^+:X L +GN &

−(B x + B x + B x + B ) sin x

u(x)v(x)dx = u(x)v (x) − u (x)v (x)

=

+u (x)v (x) − · · · + (− )n− u(n−) (x)vn (x) +(− )n u(n) (x)vn (x) dx

(−B x + (A − B )x +(A − B )x + (A − B )) sin x +(−A x + (A + B )x +(A − B )x + (A − B )) cos x

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= P (x)

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−B = A − B = A − B = A = A − B = A + B = TA = TA = " + G ! ^ L

< B = TB = TB = − TA = − TA = + =B =

-

P (x) cos(ax)dx =

)0

x sin x dx = (x − ) sin x + x( − x ) cos x + C

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-

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-

-

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x ex cos x dx, xex sin x dx.

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+

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1%

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= =

< I =

-

dx = x
2

tan x −

tan

"

-

tan x + I

tan x −

=

=

tan

=

x−

Tu =

(x −

#

x −

# x −

x

x +

+

$

2 N

x = =

tan x + I x + tan x − I

tan

=

) + x + C )>

-

2 N x

$

+C

x + x) sin(x)

2 N

V

u eu du

√ x i: & < 0= -

tan x − I

tan x( tan x − tan x +

+C

eu u − u + u − u + + C

x ln x − ln x + ln x − ln x + + C

=

V = I=6 '. &

cos(x)

(ln x) dx = =

x( tan x + ) − ln | cos x| + C

tan x dx = I =

$

x − +

Tu = ln x i: & < 0= F=6 ,: ^+Q< '. &

=

-

=

tan x dx = I = tan x − I = tan x − tan x − I

V

+ (x − x + ) cos(x) + C

tan x dx = − ln | cos x| + C

V = I=6 '. &

=

sin(x) x cos(x)dx = +

I =

-

-

tann− x dx

tann− x − In−

n−

x −

x

!

V 0= F=6 ,: ^+< '. &

tann− x tan x dx

=

2

0= F=6 ,: ^+Q< '. &

In

#

$ P ”(x) P () (x) P (x) − + − ··· a a # $ cos(ax) P () (x) P () (x) P (x) − + + − ··· + C a a a sin(ax) a

F=6 ,: ^+! '. &

2 N

√ sin x dx = u sin uudx u sin udu

− cos u(u − u + u) + sin u(u − u + ) + C √ √ − x (x − x + ) cos x √ +C + (x − x + ) sin x

V

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= − sinn− x cos x +

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a>

V+ i:

I

=

I

=

m

In−

=

=

dx = x + C

=

-

n

sin x dx =

' a

,G G.

sin x dx = − cos x + C

sinm x cosn x dx

V+ i:

Im,n

=

; A ! "

In+ =

= (m − ) sin

x cos x( − sin x) n

,n

-

n

− (m + n)Im,n

^&& <

V P dP dx

− sinm− x cosn+ x m+n m− I + m + n m− ,n

= sinm+ x cosn− x i:

na

w =6 x

& & 2j&

= (m + ) cos x sinm x cosn− x

−(n − ) sinm+ x cosn− x

n − n

a

In

x dx +C = arctan a a x +a "

V u = x +

+

arctan

,G In V+ i:

-

(x +

)

sinn x dx

; A "

+C

24 ^ ="&

m

= (m + ) sin x cosn x -

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= =

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)9

u I dx = I = + (x + x + ) a (u + a ) a u u = + + I a (u + a ) a a u + a a u u I = + + a (u + a ) a (u + a ) a u u = + a (u + a ) a (u + a ) u + . arctan + C a a a x+ x + = + (x + x + ) (x + x + ) √ √

−(n − ) sin x sinm+ x cosn− x = (m + ) sinm x cosn x

x + (x + a )n

< a = i: & TO& 3& & B &

x cos x − (m + n) sin x cos x m

n

sinm− x cosn+ x = (m − )Im−

=

x

√

+

Im,n

-

V n = L &
−(n + ) sinm x cosn x m−

(x + a )n

+

-

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<

∈ N

−nx dx (x + a )n+ x (x + a ) − a + n dx n (x + a ) (x + a )n+ dx x + n n (x + a ) (x + a )n dx −na (x + a )n+ x + nIn − na In+ (x + a )n x − (x + a )n

-

dP = (m − ) cos x sinm− x cosn+ x dx −(n + ) sin x sinm− x cosn x

= (m − ) sin

45; NA8 &4+

8 i: & 24 ^ ="& sinm x cosn x dx V P = sinm− x cosn+ x

m−

= dx

V

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-

; A "

dx (x + a )n

=

Ǳ_B & Ǳ_B \< L ! & < dv

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-

dx

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=

^&& < Im,n

sin− x cos− x

sin− x cos− x + I ,− − sin− x cos− x − sin− x cos− x

− + sin x cos x + ()I, sin x + C − − − sin x cos x sin x cos x cos x

=

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)

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cscn x dx, -

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n

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sinm+ x cosn− x

m+n

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−

9=6 =

=

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I,

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I,

=

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=

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=

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O& n< Q: L ! &

sin x + C

" "

V

secn x dx, -

-

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( )

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x e

()

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a2

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I

,

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I ,

sin x cos x + sin x cos x − + sin x cos x + I,

H%

−

sin x cos x +

m

' a

sin x cos x +

-

n

sin x cos x +

=

int(x ∗ sin(a ∗ x), x) −−−−→

sin x cos x +

−

=

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−

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sin x cos x +

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−

−

sin x cos x

x+C

sinm x cosn x dx

; A "

L T 2B5 S-= I 2Q 7[ 4-NA8 &4+ %:S + TF= I=6 %./ w0x < w x Q:

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Im,n

m+ ,n m+ m+ sinm+ x cosn+ x m + n + =− + Im,n+ m+ m+

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ǱHK G ǱHK ? )

8" & ,G Ǳ_B

K+ 1 value(Int(u ∗ tan(u2), u)) −−−−→ − cos u2 2

d '. & T%! student[intpart](I(x),u(x)) Ǳ_B & Ǳ_B @C/ L mK! < u=u(x) " i: I(x)-,G '. & . & =" ! u dv = uv − v du ! K+ k

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student[intpart](Int(x ∗ In(x), x), ln(x)) −−−−→ ln(x)x(k+1) x(k+1) − dx k+1 x(k + 1) u =

i: & Ǳ_B & Ǳ_B \< L xk ln(x) dx ,G =" ! dv = xk dx < ln(x) cd

http://webpages.iust.ac.ir/m_nadjafikhah/r1.html

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student[changevar](u=sin(x),Int((sin(x)) ∧ K+ . 3cos(x), x), u) −−−−→ u3 du u = sin x +a ++a L sin (x) cos(x) dx ,G

)F

! #$ "$ -

b

=F 8" &x V+P f (x) dx . & < + b a a V+& V+ ^&& qw" B -

b

f (x) dx = lim I P, ξ |P |→

a

& #P G d T + 4 & |P | /< " B =. P + %+& &

9!Y,O!D(7

*

[ 5 [a; b] @L & & % ! 5 & y = f (x) + i : 3 Ty = f (x) U& . & <3 @+ % =S =V+ D!3 V+P x = b s < x = a s TP x & N + / [ 5 fk 5 : " " L 8 fk % L p & & mH =V y% z =V+ ( T[a; b] @L& &

>! V! ' L p

!

N L 54 QD

P : a = x < x < · · · < xn− < xn = b

$> > & %./ n & [a; b] @L& TL: ^+$ P =%!

^+5 ,G [5 =F 8"

^+hH < O& L B< 24 ,G ' N 24 ξ + N {! ( & ^ =L {N V+G& X ξi ∈ Ii S %! ^"< =]H P ,G O& {N I P, ξ G d T" e f (ξi ) T" / f (ξi ) V+G& X ξi ∈ Ii S < T& " N+/ & =& P ,G ^+hH {N I P, ξ G d V
@L& L L: P

= {xi }ni=

Mi

=

mi

=

V+ i:

!

< %! [a; b]

sup f (x) xi− ≤ x ≤ xi

inf f (x) xi− ≤ x ≤ xi

@L& ,X =i = , , · · · , n Ii = [xi− ; xi ] V+N T24 ^ =V+P Δxi = xi − xi− . & i < V+P #P . & < + P n 24 & < |P | . & P

MV!g

>!V! /

|P | := max

Δx , Δx , · · · , Δxn

>!V!

(Q C ξ = ξ , ξ , · · · , ξn S =V+ [5 V+ [5 24 ^ Tξi ∈ Ii i P ǱL & "& & +p

(! "! h01 C ^ =I(P, ξ) := ) f (ξ )Δx =V+ ξ " iK" < P L: n

i

i

i=

& y = f (x) U& V+hS 4 ! < %! [a; b] @L& P ǱL& %! I && d n L P = {xi }i= L: P ǱL& " %: δ > Tε > ξ = {ξi }ni= + N L ( P ǱL& < |P | < δ [a; b] f I %Q ^ =I P, ξ − I < ε V+"& "

! "!

>! ! "!

)I

\](! "!

W "#:*8 ,

<#\-Q#:*8 ,

-

b

& =%! && < B f (x) dx V+ i: w a L: P L & B< δ > T (Q ε > P L + N L ( P < |P | < δ & [a, b] L P = {xi }ni= G 2DC & =|I(P, ξ) − | < ε V P & ξ = {ξi }ni= −

$ ξ

ε

ε

< I(P, ξ) − <

ε < f (ξi ) < Mi + (b − a)

Mi −

CJ h01 H h01

w =F x

' Tε > L& TMi [5 & B & B<

∈ [xi− ; xi ]

@L& & y = f (x) U& N .+ < H! {+ &

G d T"& !+H f S " Bx ="& Ii = [xi− ; xi ] w="& .++ .P .+ < ._ .P H! & [a; b] L: & %D f U& < 24 & T{+

ε (b − a)

I(P ) :=

i=

) ) ε Mi Δxi − Δxi < f (ξi )Δxi (b − a) n

<

n

i=

n )

ε Mi Δxi + (b − a)

i=

i= n )

Δxi

V TH! %+4 & B & T^&& < ε

I(P ) −

< I(P, ξ) < I(P ) +

−

ε

< I(P, ξ) − I(P ) <

T& 24 & = =

2

-

b

f (x) dx =

lim I(P ) =

|P |→

f (x) dx = lim I(P ) =

|P |→

a

^&&

|P |→

-

mi

=

Mi

=

-

I(P ) =

=

i= n ) i=

=

Mi Δxi = xi −

n )

i=

n ) i=

n ) i=

(Δxi ) ≤

n )

i=

n )

Δxi

-

b

b

f (x) dx ≤

f (x) dx. a

a

b

f (x) dx 8 & : < LO " ! - b - b B f (x) dx < f (x) dx %! ^ "& B

<

a

a

=" 8 ,G ! P N 24 ^ ="& && .b

V+ i: =

a

f (x) dx =

V+S i: w[Q A7S" T^&& =%! d } N

.b

a

f (x) dx

¯ ) = lim I(P ) = lim I(P |P |→

|P |→

L & B< δ > T (Q ε > P L & TmH V |P | < δ [a; b] L P L: P < I(P ) − < ε

,

< − I(P ) < ε

L: & ξ = {ξi }ni= + N L ( P L & V T|P | < ε P = {xi }ni= i< I(P ) <

n )

mi Δxi ≤ I(P, ξ) =

i=

n )

Mi Δxi = I(P )

i=

+ <

−ε = I(P ) − < I(P, ξ) ≤ I(P ) − < ε

(xi − xi− ) − x

|P |→

B< ,G ! ^ ^+& GG D

xi (xi − xi− )

xi− xi =

f (x) dx := lim I(P )

a

+ n )

b a

-

b

f

inf x xi− < x < xi = xi− sup x xi− < x < xi = xi

-

T"& ]KQG [a; b] & f S %! ^"< =V+ [5

G d

S 2DE

+P =f (x) = x + i: 1 ! ="& / && d ,G N < %! ]KQG [; ] & L: P : = x < x < · · · < xn = V+ i: = 24 ^ =%! [; ] &

H ! "! < CJ ! "!

<

f (x) dx := lim I(P )

w0=F x

b a

b

=−ε < I(P ) − < ε V w0=F x < w =F x & B & T -

-

a

mi Δxi

i=

{+ & b a L y = f (x)

ε

ε

n )

I(P ) ≤ I P, ξ ≤ I(P ).

f (x) dx ≤

+ <

I(P ) :=

L (Q ( P L & %! ^"< =V+ [5 V Tξ = {ξi }ni= + C

a

i=

<

Mi Δxi

i=

+ n )

n )

lim

|P |→

i=

)1

n ) i=

f (ξi )Δxi

+ =|I(P, ξ) − |

< ε

^&& <

=%! && < B

<#\-Q#:*8 ,

W "#:*8 ,

V T%! 54 I(P ) − I(P ) = =

<

n )

P& < δ

(Mi − mi )Δxi

(f (xi ) − f (xi− ))Δxi

i= n )

δ

=

(f (xi ) − f (xi− ))

=

= f (x)

≤

" i: %! : mH =%! .

U& S

" !

=%! ]KQG G d

88 @ !+H L& ^ & T%! !+H [a; b] & f $ A7S" B< δ > ' Tε > P L & T+ =%! =|f (x) − f (y)| < ε G d T|x − y| < δ < x, y ∈ [a; b] S |P | < δ & [a; b] & L: P = {xi }ni= V+ i: , C T[xi− ; x ] & f G !+H +Q & T24 ^ =%! < mi = f (zi ) S & T B< yi , zi ∈ [xi− , xi ] + =Mi = f (yi ) I(P ) − I(P ) = n ) i= n ) i=

n )

(Mi − mi )Δxi

i=

n )

n ) i=

xi− xi − #

−

n )

−

=

n− ) i=

xi −

xi− (xi − xi− ) xi

n ) i=

Dri(x) dx

=

Dri(x) dx

=

ε ) Δxi = ε b−a

lim

|P |→

lim

|P |→

n ) i= n ) i=

=%+ ]KQG [;

d @.P mH =%! ^ Qe e " + ^ S 2DE _+Q d L 7 V8 - ]KQG

@L& & y = f (x) U& + i: # ! ]KQG [a; b] & f 8 & : < LO " =%!

@ & P L& L ' ε > P ǱL& %! ^ "&

x∈Q x ∈ Q

=%+ ]KQG [, ] @L& & & (Q L: ' P V+ i: Tp ^ & = /O < S C ' /O Ii = [xi− ; xi ] @L& =%! [, ] =mi = < Mi = ^&& T B< S C ' +

-

& 6=I=> N U& ' y = f (x) S U- !

G d T"& [a; b] & y = f (x) S _+ < T"& [a; b] @L& =%! ]KQG

xi− xi

Dri(x) =

ε Δxi b−a

=%! . P& <

$

'P. + %&E 2 N

(f (yi ) − f (zi− ))Δxi

< "& [a; b] & y = f (x) U& S ! =%! ]KQG G d T"& !+H @;N P 5 _&

[a; b]

mi Δxi =

-

i=

2

S 2DE T^&& =I(P ) ≤ I(P ) .P T:X L ]KQG [; ] & f (x) = x ^&& < I(P ) = I(P ) = / ="& / && d ,G < %!

n

=

n )

i=

δ(f (b) − f (a))

T"& !+H [a; b] & y

( − ) =

i=

2

≤

=

i=

= ε/(f (b) − f (a) + )

=

I(P )

i=

i=

=

n )

=

]

Δxi = ( − ) =

Δxi =

@L& & y = Dri(x) T^&&

^ %! d L +t+H O& [5 %! ^"< T5x ]KQG @Q@ U/< =G& 5Q; L Y;! $ u & +Q ^+.P & < %! 8+8 +& w,G B< x V P j: < . U: d k / T/ ' < +A/ =%S V+P L& d & k wLH ! QG & @Q@ & .<+ . V+ & " C& ^ =V+ v;5 w,G ^ : T5x +GQG < 8 [a; b] @L& & y

U& S ! ! =%! ]KQG G d T"&

= f (x)

=%! < 54 [a; b] @L& & f + i: A7S" P = {xi }ni= + i: w=%! & TQ<_ %Q 2DEx f $ T|P | < δ S T24 ^ =%! [a; b] & L: )J

W "#:*8 ,

<#\-Q#:*8 ,

L n" & P L& = I

=

−

+ i:

;

,···,n

(Ii

i=

m

P L & T /i ∈ Ii n< & T24 ^ ^+t.P =i > n P L & /i ∈ In+
,X)

=

≤

n ≥

/ε Tm

m(n − )

m

+

+

n )

≥ ε/( n+ )

n

+

m

n

−

/(k −

))

i: &

- /(n+) n - /n )

=

=

f (x) dx

lim

n→∞

−

n+

ε >

|f (x) − f ()| =

=

= = x = f (x )

P L& G d Tx = S √ V T|x − | < δ = ε f (x) =

=%+

x = S x = S ]KQG [; ]

x ∈ QS x ∈ QS

x <δ =ε

2 N

@L& P & f (x) = [x] Y+34 Ǳ_B U& +P =%! ]KQG TP < & +G !+H 24 ^ TI = [a; b] V+ i: = 5 TI @L& Y+34 N L DC I & y = f (x)

>

S S P 24 ^ i P L&

(n − n )ε

;i +

= ε & %! @+A/ && T+ =(n − n ) × (n − n )ε =%! ]KQG I & y = f (x) U& )= =F f (x) = [x] . _+ ; Q@ ^ =%! ]KQG [a; b] & F= =F @+ && ]Q < %! 8

U& +P w> ,e f (x) =

@L& &

T%! ]KQG [; ] @L& & f (x) = x U& + %&E w> =+ D!3 d ,G N mK!

(n − n )ε

&& P L& ^ ,X . < I ∩ Z ⊆ In ∪ · · · ∪ In G d

U& + %&E w0

[sin ( /x)]

x

_& B .P y = f (x) $ T )= =F @+A/ && T+ =%+ ]KQG [; ] & + < %! !+H x =

Ii := i −

& f (x) = x − U& + %&E I= =F L ! & w & d ,G N mK! T%! ]KQG [; ] @L& =+ D!3 9= =F L !

f (x) =

P <

< "& (Q ε

!

x

n→∞

I ∩ Z = {n , n + , · · · , n }

n dx n→∞ /(n+) n + i= n ) lim n→∞ (n + )(n + ) i= lim

= f (x)

≤

(Ii

n→∞

& y

k −

k=

f (x) dx = lim =

n )

k=

-

+P 24 ^ =%+ x y = f (x) 24 ^ Tx = V+ i: = B< {xn }∞ n= S C L QD L T%! !+H Q w-$x %! G.P x &

mn(n + )

,X) < ε < ε V Tm ≥ n i=

@+A/ s" T5 =

,]KQG [; ]

n→∞

lim f (xn ) = lim

+ ··· + m m +

<

/(n − ) − /n m

1 !

x ∈ QS x ∈ QS

x

f (x) =

···

+

=i = n+ )

L . P Ii ,X . TIi ⊆ [a; b] " %: $ Ii 2 ="& [a; b] & f G !+H N . " Ii < "

,

In+

In

m

;

m /n − /(n + ) − ; + = n m n

/n − /(n + ) = ; − n m

ε

−

=

I

≤ f (x) ≤

.

n n+

< x ≤ n S x = S

< %! ]KQG [; ] @L& & Tn ∈ N /n 8" & N f U& %! ^"< = =

)

f (x) dx =

/

n+

W "#:*8

! ,

-

=+ D!3 V 9= =F ,: ^+Q< '. & =

< %! ]KQG [; π] &

-

]KQG (Q @ & @L& P & L U& +P w6 %! x∈ / Q S f (x) = /n x = m/n ∈ Q, (m, n) = S

f (x) dx =

−

=

lim

−

n→∞ lim

n

n

− +i

f

i=

n )

n

))

n

n→∞ lim

=

(x + ) dx

− (−

n→∞

=

W "#:*8 ,

i= n )

i

− +

n

n→∞ lim

+

i=

=

@L& & f (x)

+

%! ]KQG [;

f (x) =

i=

U& +P T

2 N

U& & ]KQG i: & =+ D!3 L y = f (x) ,G T[; ] V 9= =F ,: ^+< '. & = -

f (x) dx = =

= =

− n

lim

n→∞

lim

n→∞

lim

n→∞

i=

n− )

n

i=

n

i n

− n

+i

f

n− )

= lim

n→∞

n(n − )(n − )

n =

-

)

x dx,

)

x dx,

)

-

- -

i=

(x − ) dx, (x − x + ) dx.

-

n−

(− )

f (x) dx

&&

-

f (x) dx

< a

[5 G d Ta = b S T
-

a

*

!

-

b

n=

(− )n+ /n i: &

/(n + ) < x ≤ /n S x = S

w1

@L& &

[a; b]

y = f (x)

U& S

!

G d T"& ]KQG

b−a ) lim f n→∞ n n

b

=

i=

a

b−a lim n→∞ n

n− ) i=

b−a a+i n

b−a f a+i n

a

f (x) dx := − f (x) dx b - b = f (x) dx = V+ a

3∞

=

4

< %! ]KQG [; ] @L& & =%! π / −

L @+A/ T"& " ^+.A ,G B< 4 ^ 2DE =P h ,G @D!3 & ! \< = @C & ^. C & +A/

f (x) dx

V+ [5 Tb

wI

i: T j: H TmH ^ L &&) ! = B " n; QG . %! ^ &

-

^ !D(7

/n = π / 8 i: & /(n + ) < x ≤ /n S x = S

=%! ' L S_& 5+DX C ' a + i: wJ U& +P −n −n −n a <x≤a S a f (x) = x = S

i

T " QG B< i: & ! ! + D!3 9= =F '.8& d L ' P N

L U& +P wF x = S x = S

i=

U& +P Tln =

&&

T%! C. @D!3 +A/ ^ & ^ .C Zj 2CfX +.8 & L+ d :5 & DQ

=w" B 6=>=F &x ,G

)

3∞

x dx

] @L& & /x − [ /x]

/n f (x) = - && f (x) dx
f (x) = n− )

i

= x

-

n

n )

n

)

n(n + )

n→∞ lim n + lim n→∞ n n + =

=

− (−

+ %&E w9 =%! 4 && d ,G f (x) = sin x

2

U& & ]KQG i: & ! − L y = f (x) ,G T[− ; ] @L& & f (x) = x +

W "#:*8 ,

W "#:*8

= M

n )

C < y = g(x) < y = f (x) U& L & ! V Tα < wa < b < c x c Tb Ta (Q

Δxi = M (b − a)

i=

=m(b−a) ≤

2

-

b

a

-

f (x) dx S 2DE & 24 &

-

f (x) dx, m ≤ ≤ M

a

a

B< c C T"& !+H g S -

-

b

)

a

a

T"&

f (x)g(x) dx = f (b)

a

b

g(x) dx c

2

L& d & y = g(x) < ]KQG [a; b] & y = f (x) S < a ≤ c ≤ b B< c C G d T"& 8 -

-

b

f (x)g(x) dx = g(a) a

-

c

b

f (x) dx + g(b) a

f (x) dx c

& f V+ i: T3& %+7 L " ! <& w x A7S" / {+ & m < M S T24 ^ =%! [a; b] x P L & G d T"& [a; b] & g U& N e < && mH =mf (x) ≤ f (x)g(x) ≤ M f (x) n ≤ g(x) ≤ M V >=0=F @+A/ L wFx %./ -

-

b

f (x) dx ≤

m a

m ^+& μ :=

f (x)g(x) dx ≤ M a

.

b a

-

b

f (x) dx +

&

f

-

g(x) dx, a

c

f (x) dx, b

.++ ._ {+ &

m(b − a) ≤

m

<

S w6

G d

M

b

f (x) dx ≤ M (b − a)

G d T-f (x) ≤ g(x) - x ∈ [a; b] P L & S wF b

a

g(x) dx a

& _+ f g G d T"& ]KQG [a; b] & g < f S wI =%! ]KQG

[a; b]

! < V+ 2DE w6x < w0x 8 A7S" V8 S %! ^"< =VK! & ^. C &

! w x V8 L mK! < V+ 2DE + %Q & w0x [a; b] & g < f V+ i: =S 2DE _+ − %Q TV+

δ > C T^&& =%! (Q ε > < ]KQG |P | < δ & [a; b] L P L: P L & " %: $ δ > T& 24 & =I f (P ) − I f (P ) < ε/ V |P | < δ & [a; b] L P L: P L & " %: $ L: P L & T24 ^ =I g (P ) − I g (P ) < ε/ V V |P | < δ := min{δ , δ } & [a; b] L P

I f +g (P ) − I f +g (P ) = sup f (x) + g(x) a ≤ x ≤ b

− inf f (x) + g(x) a ≤ x ≤ b

≤ sup f (x) a ≤ x ≤ b + sup g(x) a ≤ x ≤ b

− inf f (x) a ≤ x ≤ b − inf g(x) a ≤ x ≤ b = I f (P ) − I f (P ) + I g (P ) − I g (P ) ε ε + =ε ≤

b

f (x) dx a

. b f (x)g(x) dx ÷ a f (x) dx C ^&&

= / M < G !+H L T !+H U& & + N @+A/ && w0x B< $ c ∈ [a; b] V+S + [a; b] & g =μ = g(c) < %! | ^ @74 L V8 ^ 2DE w>x =+S / 3& _+Q d

b

f (x) dx ≤

f (x) dx a

b

-

b

a

[a; b]

b

f (x) dx ±

b b ) f (x) dx ≤ |f (x)| dx. a a

c

V TQ<_ f %Q -

-

c

-

b

a

f (x) dx =

2

f (x)g(x) dx = g(a)

-

a

k+ < %De L& d & g < ]KQG [a; b] & g < f S < a ≤ c ≤ b B< c C G d T"& Q<_ b

b

-

b

-

f (x) dx, a

a

a

-

b

(f (x) ± g(x)) dx =

f (x) dx, a ≤ c ≤ b

f (x)g(x) dx = g(c) a

2

-

)

b

f (x)g(x) dx =

αf (x) dx = α a

2

b

-

b

)

E- )4 < y = f (x) S ! < ._ {+ & m < M T]KQG [a; b] & y = g(x)

G d TP %fC ++a [a; b] & f < "& [a; b] & g .++ B< C

-

! ,

& L: P -

=" 2DE w0x {+ ^ & < S " B %! : w6x 2DE &

G d T"& [a; b]

b

f (x) dx ≤ I(P ) = a

0

n ) i=

Mi Δxi ≤

n ) i=

M Δxi

W "#:*8

! ,

W "#:*8 ,

x > P L & +P

x < ln(x + ) < x x+ /(x + ) V+ i: >=0=F L w0x

Tf (x) =

-

b

M

inf

=

sup -

x

dt t+

t+

t+

L w9x < a ≤ c ≤ b B< c C T^&& =V+ !

-

-

b

a

-

-

b

-

)

-

x

)

cos x dx +

)

dx

-

−x

-

π

-

b

-

π/

)

=

≤

-

π

)

)

f (a) + f (b)

≤

=

-

π

-

-

sin x dx

[x] dx

e−x dx

)

-

π

+x

−

dx ≤

2 N

- / =

-

=

+

e−x cos x dx

=

+

e−x cos x dx

>

-

-

sin x dx =

N V =0=F @+A/ L w>x - /

[x] dx

/

-

[x] dx

dx

/

/

dx

− + − − + − =

2 N

[x] dx

dx +

π

[x] dx +

dx +

- /

sin x dx

+ T

[x] dx +

- /

π

+c

- /

a

)

≤x≤

-

+

f (x) dx ≤ (b − a)f (b)

π

√

%./ & B & =+ D!3

b

-

=

sin x = +x

+ ( L QG L ' P %fC π/

+x

sin x dx = +x

G d T"& 3 < 54 f S _+ < (b − a)

| −

=

Tg(x) = /( + x ) Tf (x) = sin x + i: >=0=F @+A/ L ,< %./ M&; T^&& =b = π < a = < ≤ c ≤ π %P c '

f (a) + f (b)

a

+

√

2_

f (x) dx ≤ (b − a)

min

≤x≤

+x

+

=

| −

V 0=0=F @+A/ L wFx %.: L ! & T+

-

b

max

=

< 54 [a; b] @L& & y = f (x) U& S + %&E

G d T"& 5N -

2

[ sin x] dx

2,

=%! . P& <

+ x V+ i: ! 24 ^ =b = < a = −

M

/ b 0 b b 0 1 f (x)g(x) dx ≤ f (x) dx g (x) dx a a a

(b − a)f (a) ≤

=

=

& g < f S + %&E w_" ` 8 +& < x

G d T"& ]KQG [a; b]

c

2

m

f (x) dx

a

Tf (x)

b

f (x) dx −

a

dx

x + dx x + - ) ex dx

sin x dx x π/ - x dx ) + x

-

c

f (x)g(x) dx = g(a)

+ D!3 L QG L ' P DN N )

f (x) dx a

-

π

f (x) dx =

= (g(a) − g(b))

& L QG L ' P N ! + D!3 ,G @" S Z < [5 L ! )

b

- ac

a

x

f (x) dx a

f (x)g(x) dx − g(b)

= = λ dt < m ≤ λ ≤ M %P λ ]Q < V8 ^&& = /x + ≤ λ ≤ < ln(x + ) = λx 5 =" + p

c

f (x)(g(x) − g(b)) dx = (g(a) − g(b))

< t ≤ x = x+ < t ≤ x =

=

f (x)(g(x) − g(b)) dx ,G w>x V8

a

%./ = V T24 ^ =b = x < a = Tg(x) =

m

-

2 N

W "#:*8 ,

Z*% O ` W58 CD ,

|F (y) − F (c)|

= = =

- y - c f (x) dx − f (x) dx a -ay - a f (x) dx + f (x) dx a c - y f (x) dx

G d ) )

c

≤ =

)

M |y − c| ≤ M δ Mε <ε M+

)

=%! !+H c F + < & T^&& ="& !+H c ∈ (a; b) f V+ i: , y ∈ [a; b] S B< δ > ' T (Q ε > L S T24 ^ =|f (y) − f (x)| < ε G d T|y − c| < δ <

G d Tc < y < c + δ F (y) − F (c) = − f (c) y−c F (y) − F (c) − (y − c)f (c) = y−c y = f (x) dx − (y − c)f (c) y − c c - y y = f (x) dx − f (c) dx y − c c c y = f (x) − f (c) dx y − c c - y ≤ f (x) − f (c) dx y−c c - y ε dx < y−c c =

y−c

< y < c

%Q & 24 & =%! .

@+Q< U& ' F (x) S H5;[ c /- !

G d T[a; b] ⊆ (α; β) < "& (α; β) @L& & f (x) U& -

f (x) dx = F (b) − F (a)

a

<

n→∞

π/

- lim

n→∞

sinn x dx = xn dx = +x

- lim

→+

-

bε

lim

ε→+

dx

x +

aε

=

f (x) dx = f () ln x

b a

=+ 2DE >=0=F @+A/ L w x %./ w J =+ 2DE >=0=F @+A/ L w>x %./ w0) .Px =+ 2DE >=0=F @+A/ L w9x %./ w0 " i: S + ! 8 ^ L

G d Tf − := (|f | − f )/ < f + := (|f | + f )/ w=|f | = f + + f − < f = f + − f − =+ 2DE >=0=F @+A/ L wFx %./ w00 .Px =+ 2DE >=0=F @+A/ L wIx %./ w0> _+ f G d T"& ]KQG f S +P & wf g = (f + g) − f − g / mK! < %P

X( $L _ ^4 7 @A

b F (x) a

. & O& < %! %.! 2DC =P W.

=%! [a; b] @L& & L: P = {xi }ni= + i: A7S" TR SO @+A/ && i = , · · · n P L & T24 ^ < xi− < ti < xi B< ti C F (xi ) − F (xi− ) = F (ti )(xi − xi− )

*

< ^+5 ,G ^+& D TW(& ^ L vP =%! ^+5 ,G /J FG ^ Y* & J &/K

b

a

b F (x)

lim

(y − c)ε = ε

P& < S 2DE c − δ 2

+ %&E L < ' P

-

P L& < %! !+H [a; b] @L& & y =F (x) = T%!

(a; b)

&

-

!

= f (x) x

U& + i:

f (t) dt a

x ∈ [a; b]

@+Q< U& ' F (x) 24 ^ =F (x) = f (x) x ∈ (a; b) P L& 5

f (x)

T%! ]KQG f $ =ε > < c ∈ [a; b] V+S A7S" B< S & M C ^&& < "& mH V+ i: =|f (x)| ≤ M x ∈ [a; b] P L & Ty ∈ [a; b] < |y − c| < δ S T24 ^ =δ = ε/(M + ) 9

Z*% O ` W58 CD ,

x

W "#:*8 ,

V . TmH

& %D b(x) < a(x) Tf (x, t) U& S ! 24 ^ T"& ]KN d dx

-

b(x)

f (x, t) dt

n ) F (xi ) − F (xi− )

F (b) − F (a) =

i=

= b (x) f (x, b(x))

a(x)

−a (x) f (x, a(x)) +

-

n )

= b(x)

i=

∂ f (x, t) dt ∂x

a(x)

n )

=

F (ti )(xi − xi− ) f (ti )Δxi

i=

∂ f (x, t)

∂x =%! t &

%&E i: & < x & %D f (x, t) M

2

V 9=>=F '. & d dx

-

= =

!

G d Tξ = {ti }ni= " i: S T^&& F (b) − F (a) = L(P, ξ)

+S & T =I(P ) ≤ F (b) − F (a) ≤ I(P ) + V+S + T|P | → & ^+:X L

x

F (b) − F (a)

sin(xt) dt = x

x sin(xx

-

x x

)+

t cos(tx) dx x

2 N

a

2 -

T

cos x dx = sin x + C -

- x √ d ln(x + t) dt = √ ln x + x dx /x x - √x − dt − ln x + + x x /x x + t

√ x √ ln (x + x) ln x + x √ + = + ln(x + t) x x /x √ ln x + x + ln x + = + √ − x x x

=

V T

2 N

V , +HP @C/ < 9=>=F '. & P =

-

-

V T

=

dx

dx

/

e−x dx = −e−x + C

e−x dx

-

x sin x lim x→ x lim sin x = x→ x

−x

−x

=

−e−x

=

−

2 N

−e

.x

(arctan t) dt lim x→ x x +

P

=

=

lim

x→

=

arcsin( ) − arcsin

/

π

lim

x→

(arctan x) = x

6

−

π

=

x cos x dx

e

2 N

& B &

=

π

2 N

& B &

V

=

π x sin x + cos x

=

(− ) − ( ) = −

π

−

arcsin x

x cos x dx = x sin x + cos x + C -

(arctan x) + x x +

√x x +

+ e =

= arcsin(x) + C

-

T

2 N

& B &

=

V , +HP @C/ < 9=>=F '. &

V

π/ sin x π − sin = sin

=

-

!

cos x dx

=%! . P& <

2

& B &

π/

√

√ sin( t) dt lim x→ x

f (x) dx

t cos(xt) dt

-

b

=

V , +HP @C/ < 9=>=F '. &

. x

lim I(P )

|P |→

-

x

) − sin(xx) +

x sin(x ) − sin(x

=

W "#:*8 ,

Z*% O ` W58 CD ,

n+n n ∞ ) − ln + +k n

& < [;

· · · + ln =

lim

n→∞

k=

- = =

− n

lim

n→∞

=

- x x ln( + x) − -

−

x+ dx

x+

D!3 L QG L ' P N

) )

−/

-

sinh

sinh

dx

)

−x dx +x

dx x + x+ − -

) arctan x dx

)

) )

| − x| dx

-

)

−

) ) ) ) )

n→∞

n→∞

√

-

x dx

n→∞

=

=

dx x + / π

)

+ ···+ n

+

n n

+

k=

k n

2# N

Tf (x) = xp Tp > i: & V 9= =F

+ p + · · · + np = np+ p p n p lim + + ··· + n→∞ n n n n n p ) k lim n→∞ n n k= p n − ) − lim +k n n→∞ n k= - xp dx p

lim

sin x dx π/ √

$

n − ) n + ( + k − n ) k=

dx = ln | + x| = ln +x

a = < b =

- π/ -

n

lim

+

+

=

n )

lim

=

+ + n

n

-

! !

- √

lim

=

=

dx cos x +

=

-

dx x +x - b - x d ) sin x t dt ) d sin t dt dx a dx a - x x dt d d ) ln(x + t ) dt ) dx x dx x x + t arctan(xt) dt ) d dx t .x

+ + ···+ n+ n+n #

n+ n→∞

=

) = /e ^&&

/(x + )

dx

= = exp( ln −

2+ N

i: & V 9= =F L !

=

= ln − x − ln( + x) = ln −

)

@L& & f (x)

ln( + x) dx

= ln −

- /

]

+ D!3 L < L ' P N

.x (arctan t) dt cos t dt lim ) lim x x→ x→ x x + . x t - x e dt sin t dt lim . x t

) lim t x→ x→ x e dt . sin x √ √ n tan t dt n! lim . tan x √ ) n→∞ lim n x→ sin t dt + ···+ n − lim + n→∞ n n n n n n lim + + ···+ n→∞ n + n + n +n

=

lim

n→∞

n

xp+ p+

= ln x

=

p+

n (n + )(n + ) · · · (n + n)

2% N

N

=+ D!3 V+ i: p ^ & =

an :=

Ty

n

n (n + )(n + ) · · · (n + n)

G !+H +Q & 24 ^ =

< V

:= lim an n→∞

lim an = lim ln(an ) n→∞ n→∞

(ln(n + ) + ln(n + ) + · · · lim n→∞ n

ln = =

=

=

F

ln

+ ln(n + n)) − ln n

lim ln(n + ) + ln(n + ) + · · · n→∞ n · · · + ln(n + n) − n ln n

n + n+ lim ln + ln + ··· n→∞ n n n

W "#:*8 7 #X* #X5 ,

W "#:*8 ,

G(g(t)) < F (t) ^&&

= G(g(t)) = f (g(t)) g (t) + B< C C T^&& = && M (a; b) & L =F (t) = G(g(t)) + C (Q t ∈ [a; b] P L &

V t = a L & :X

)

f (g(t)) g (t) dt = .α G(g(a)) = G(α) = α f (u) du a

=%! . P& < C = ^&&

2

2

V+ p f Tt = ln x i: & x t

^&& = ln x

-

dx x

= =

^&& = √

-

−

√

−x

=

=

π/ −π/

t+

dx =

π/

−π/

=

−π/

= tan (x/)

arctan

arctan

lim

exp

i: & x t

n

t

=

π

√

√

√

)

n→∞

)

n→∞

lim

π

+ ···

n

+

+

+ ···+

n + n

n

2#

h h .f a + ···f a + n n n n - a+h ln f (x) dx a

n

+

+ .

√

···

n

+

n n

+ 2DE L < L ' P

√+ · · · +

√

n

=

+ + ··· n − π + = n − (n − ) n

+ n −

^ !D(7 6 ! V( ! V4

-

b

β

f (x) dx =

f g(x) g (t) dt

α

M + L @C/ &&

= sin t

+

*

%! !+H [a; b] & y = f (x) + i: ! M < g([α; β]) = [a; b] & %! 5& y = g(x) < 24 ^ =(α; β) & !+H -

2 N

$

n n

lim

√ - π/ - arcsin x t × cos t sin t dt dx = x( − x) sin t( − sin t) π/ - π/ π = = t dt = t

n

a

√

h

+

2 N π/

+ sin

N +P ,e C & mK! %! exp ((π − )/)

n→∞

2 N

" p f Tt = arcsin( x) i: & = xt π/ < dx = sin t cos t dt Tx

- - dt dx dt +t = = −t cos x + t + +t + √ √ √

=

=

π + √

^&& π/

+

n

n→∞

( cos t) ( cos t) dt

π/ sin(t) =

n

( + cos(t)) dt

V+ p f Tt -

ln

√

-

=

π

lim n f a +

sin t i: & √ − < dx = cos t dt −π/ π/

=

x t

ln

t

n

&& L

t dt

lim

n→∞

ln

V+ p f Tx

dx x

n→∞

sin

]KQG < %De [a; a + h] @L& & y = f (x) + i: +P 24 ^ T%!

!

< dt = ln

lim

(n − )π · · · + sin n #

.b

F (a) =

-

)

f (g(x)) = f (g(x))g (x) -

t

A7S"

=F (t) = f (g(x))g (x) dx V+ [5 < t ∈ [a; b] V+S a =F (t) = f-(g(t)) g (t) t ∈ (a; b) P L & T24 ^ x =G(x) := f (t) dt x ∈ [α; β] P L & V+ i: , α i: , =G (x) = f (x) [α; β] P L & T24 ^ 24 ^ Tx = g(t) V+

-

g(t)

G(g(t)) =

f (u) du α

I

W "#:*8 ,

=

π/

cos θ dθ (sin θ + cos θ)

- π/

=

I

cos(π/ − θ) d (π/ − θ) (sin(π/ − θ) + cos(π/ − θ))

= -

2+ N

" p f Tt = π/ − x i: & ^&& < xt π/ π/ Tdt = − dx

a cos θdθ (a sin θ + a cos θ)

=

π/

W "#:*8 7 #X* #X5 ,

π/

=

π/

=

^&& -

I

=

π/

=

π/

-

I

dθ sin θ + cos θ

= =

V Tt = tan (θ/) i: & mH - I

=

=

= =

I

) )

e

cos bx dx = #π

= = =

=

Re #

=

Re

= =

=

e

a + bi

e

= eax Re(ebix ) dx $

dx

(a+bi)x

π

$

a − bi (a+bi) π Re (e − ) a +b

Re (a − bi)(e πa cos(πb) a +b − + e πa sin(πb)i) e

aπ

a +b

=

π

(a+bi)x

π/

+ sin x

π/

cos x + sin x

=

2, N

ax

dx +

dx

dx cos x + sin x π/ π dx = x =

-

π

cos x + sin x

2# N

s7 ( C L TL ,G N @D!3 & V+ ! -

dt

=I = π/ ^&& < T dx = − dt " p f Tt = π − x i: & π ^&& < xt π

−t + t +

dt √ ( ) − (t − ) √ √ √ − (t − ln + (t − √ ln √ +

=

π/

dt

+t t −t +t + +t - dt

-

-

π/

cos x + sin x cos x + sin x

=

cos t + sin t sin x

cos x

π/

-

sin t

π/

=

sin θ + cos θ dθ (sin θ + cos θ)

dx cos x + sin x sin t (− dt) sin t + cos t

-

=

sin θ dθ (sin θ + cos θ)

cos x

π/

π

x sin x dx + cos x

- (π − t) sin t (− dt) + (− cos t) π - π (π − t) sin t dt + cos t - π π sin t dt t sin t π − dt - + cos t - + cos t π π d(cos t) x sin x −π − dx + cos t + cos x

π −π arctan(cos t) − I

−π ×

−π

+π×

π

π

−I = −I =I = π / I = π / ^&& - π =%! 4 sin m x cos n+ x dx + %&E 8 & T"& I " ,G V+ i: = I = −I S 2DE %! : I = =

2% N

- I

=

sin

m

(π − x) cos

-ππ =

a(cos(πb) − ) + b sin(πb)

I =

1

(sin

m

x)(− cos

n+

n+

(π − x)d(π − x)

x) dx = −I

V x = a sin θ i: & a

a dx (x + a − x )

2* N

dx

W "#:*8 7 ǱZ= 4 ǱZ= ',

W "#:*8 ,

+P T"& 4 [Q( C b < a + i: w0I

-

π/

dx

(a sin x + b cos x)

D!3 L QG L ' P N

π(a + b ) a b

=

-

dx (x + ) x + - a ) x a − x dx

)

)

^ !D(7 6 ǱX: 3 ǱX:

%*

@L& " L& @L& ' & v(x) < u(x) S ! 24 ^ T"& !+H M [a; b] -

b u(x)dv(x) = u(x)v(x) −

b

a

a

-

& V8

u dv = uv −

v du

v(x)du(x)

= ln x

A7S"

V+ i: 1 ! + < v = x < du = dx/x ^&&

ln x dx

=

-

<

= =

^&& Tdv

= cos x dx - I

= =

π/ x sin x −

π/ π + cos x = −

)

2 N

< u = ex + i: + < v = sin x < du = ex dx

= sin x dx

< u = ex i: & k ^&& < v = − cos x

$

- x x −e cos x − −e cos x dx

=

e sin( ) −

=

e sin( ) + e cos( ) −

TI = e(sin( I=

e

)

a−x dx a+x

π

-

)

(x sin x) dx

π/

dx − sin x

π/

a

-

)

+ 2DE L < L ' P

-

f (sin x, cos x) dx =

π

-

-

π/

-

π

xf (sin x) dx =

b

f (x) dx = (b − a) a

-

-

π

-

=

- x e sin x − ex sin x dx

-

f (cos x, sin x) dx

π

-

f (sin x) dx f (a + (b − a)x) dx

π/

f (sin x) dx =

f (cos x) dx −π/

-

t

π/

f (x)g(t − x) dx =

t

f (t − x)g(x) dx

+P 24 ^ T &E C q < p + i: w00

# I

)

sin x dx

ex cos x dx

T du = ex dx V Tdv

)

dx

π/

π

)

2 N

V+ i: + < v = sin x < du =

u = x

π/

dx x

ln − x = ln −

= cos x dx

x cos x dx

x

=

^&& Tdv

x ln x −

x

dx

dx dx

) ) x+ a −x x( + x ) - π - dx x sin x ) ) dx cos x + sin x − x + - π dx ) ( + cos x)( + cos x) - π/ - π sin(x) dx ) ) cos(mx) sin(nx) dx cos x + sin x −π π

=%!

= dx

a

-

a

8 & B &

-

ex −

sinh x dx

)

b

2

Tdv

)

ln

ln

+

- ) x − x dx

-

-

!

= =

) + cos( )) −

(sin( ) + cos( )) −

+

( − xp )/q dx =

-%! s7l

−I

-

-

-

-

( − xq )/p dx

- − dx = = − @D!3

x − − x

w0>

+P T < b < a + i: w09

π

π dx = a + b cos x a −b

+P T < b < a + i: w06

π

dx aπ = (a + b cos x) (a − b )/

+P T"& 4 [Q( C b < a + i: w0F

-

π/

J

dx a sin x + b cos x

=

π ab

W "#:*8 ,

W "#:*8 7 ǱZ= 4 ǱZ= ',

(n − n −

= a

== =

)

(n − n −

= a

)

n

a

n + n

a

n +

| ! p ,: T ! <

-

b

<

u

a − x dx

N =

I

I

···a

-

]SB L mH < =S

ln x dx =

I =

-

e

-

ln x dx

= =

= b a −b −

−x

b

x b = b a − b − I + a arcsin a

b

b a a −b + arcsin

π/

cos

n

x dx =

a −x

In

= =

π/

b a

= (n − )

2% N π

= (n − )

= a In− −

n

-

−

te

t

dt

tdet

te

t

−

-

t

e dt

2+ N

In

n :X L =In = a n + -

I

= =

In−

cosn− x( − cos x) dx

In

0)

=

a

n ≥ P L& ^&&

a

(a − x ) dx a x = a x−

x sin x dx

π/

-

π/

-

(a − x )n dx = - a - a = a (a − x )n− dx − x (a − x )n− dx - a udv = a In− − a a (a − x )n (a − x )n dx + = a In− − (x) −n −n

−(n − ) sin x cosn− x dx cos

t e

t

a

n−

× × × · · · × (n) a n+ × × × · · · × (n + ) ="& In && $ %.! ,G V+ i: Tp ^ & In Tdv = (a − x )n− x dx < u = x i: & 24 ^ & %! &&

π/

-

t det

a

-

(n − )(n − ) · · · × (n)(n − ) · · · ×

et dt

(a − x )n dx =

π/ n− x sin x cos −

t

n 5+DX C P ǱL& V+ %&E V+P -

cosn− x cos x dx

-

-

-

%! In $ %.! ,G V+ i: p ^ & = ]Q T dv = cos x dx < u = cosn− x < -

−

= e − et = − e

dx

V TI { & Q5

n 5+DX C P ǱL& + %&E -

= −e +

L mH I = b a − b − I + a arcsin (b/a) +

I=

t e t

= −e +

dx a −x

a +a −x dx a −x - b = b a −b − a − x dx + a

^

= e− b

t et dt

t det

= e−

a − x dx

b = x a −x −

-

e

V ^ :S Ǳ_B & Ǳ_B & ! &

2# N

≤ b ≤ a

e

a −x

b

x t

+ <

+ i: =+ D!3 = a − x V+ i: p ^ & = + < v = x Tdu = − dx ^&& Tdv = dx

-

2 N

T dx = et dt ^&& =x = et V t = ln x i: &

In−

n n +

In−

a +

<#:8Q >$ 7 ,,

W "#:*8 ,

c) ln ab = b ln a

=In

d) ln (a/b) = ln a − ln b.

2

< < e < B< e C +P e

@L& & y = ln x U& G !+H L .Px =ln e = w=+ ! [; ]

I

=

2

=logb a = (logc a)/(logc b) +P

f (x) =

x

=

2g

=

T]KQG 5& f (x) + i: w00 V+ -[5 n ≥ P L & <

f (x) dx

^&& +

= (n − )In− − (n − )In

n − n I (n−) n − n − I = ··· n (n − ) (n− ) n − n − · · · × I n (n − ) × - π/ (n − )(n − ) · · · × × (n)(n − ) · · · × × (n − )(n − ) · · · × π (n)(n − ) · · · ×

=

n

wax & Z logb a := (ln a)/(ln b) [5 & f =+ %&E logb x & wex -

In

n− n In−

=

=

=%! . P& <

x

fn (x) =

fn− (x) dx

n P L &- +P T24 ^ x

fn (x) =

(n − )!

!

(x − t)n− f (t) dt

)

n, m ∈ N P L & + %&E w0> -

( − x)n xm dx =

)

m!n! (m + n + )!

)

w=+ ! f (x) = xm & 00 ^. L .Px

)

9! D7O ;# 6

**

)

! !

@L& & %! +. y = f (x) + i: [x; x + dx] '$ @5;/ + =%! " UL [a; b] s! [a; b] @L& S mH =%! df = f (x) dx DN %+:g V+ V+N %./ n & xn− < . . . Tx Tx

-

)

)

i=

)

b

f (x) dx a

)

2

@ < L 4 + V D!3 T < a < b P y 3 ,

! !

)

x dx

√ x dx

)

-

-

π

ex sin x dx

xm ( − x)n dx

)

π

x sinn x dx

-

)

arctan

ln x dx

sinn x sin(nx) dx

-

-

)

-

e

(arccos x) dx

n

π

-

-

)

x ln( + x ) dx

& & P && kN+/ [a; b] |P |→

n+

arctan

)

dx

-

)

i=

f (xi )Δxi =

cos

-

)

lim

x

π/

)

[a; b]

-

x e

)

x cos x dx

-

@L& %+:g T24 ^ n )

@L& %+:g T. ^ L +S L mH = f (xi )Δxi n )

π/

-

P : a = x < x < · · · < xn− < xn = b

& %! && kDN

dx

π/

x dx

π/

-

π

-

cosn x cos nx dx

π

-

cosn x dx π

-

√ x− dx

sin x

e−ax cos

n

x dx

π/

cosn x sinm x dx

b

(x − a)m (b − x)n dx a

-

a

xm

-

a

ax − x

(a + x )(

dx

n+)/

dx

24 & ln : (; +∞) → R 5+DX V GQ U& w0 . & k:4 T24 ^ =V+ [5 ln x := x dx/x %De C b < a S +P [5 ^ L !

G d T"&

(x − b) + y = a

=+

@L& P x 3 & ] @ j %! ^"< =

L 5;/ < L 4 VB V =%! [b − a; b + a] =%! [x; x+Δx] @L& && P x 3 & d j V+&

a) ln

0

=

b) ln(ab) = ln a + ln b

W "#:*8 ,

<#:8Q >$ 7 ,,

`0=F 8" &x =%! v = xw && d N %C! mH T%! dK = V dm && 5;/ ^ * T+ w=" B && d * T^&& =dK = πx w δ dx 5 T%! & %! -

K=

-

r

dK = πw hδ

r

&x TV+ & y < Δx @C/ & 7+; 5;/ ^ S 4 @ ! V 24 ^ w=" B [Q`0=F 8" & & P && +; ^ < L dV

π x dx = w hδr

(π(x + Δx) − πx ) × y π(xΔx + Δx ) a − (x − b) πx a − (x − b) dx

=

= ≈

& %! && 4 VB V T^&& -

-

b+a

V =

b+a

dV = b−a

b−a

πx

a − (x − b) dx

V x − b = a sin t i: & -

' Y;! L ./ @D!3 w[Q >=F 8" U& < . ^+& <3 @+ % w

−π/

2 N

x + y = ay @ ! Y;! L ./ % =+ D!3 %! " & x + y + z = a @ s! V+! <&< 8" p Y;! L $ ' =

4 5" < P y 3 ^+& <L O V+ i: =V

24 ^ ="& p @ (, a/) @;N L & \j Y;! L " B @5;/ = ≤ θ ≤ π &x =V+S p %! θ + dθ θ c/ xOy @34 @;N 2j ( T24 ^ w=" B [Q`>=F 8" a a a sin θ, + cos θ && θ & +p T]Q q%! & && ;N d ! z=

π/

V = = =

πa πa

a − a sin t (a cos t) dt

π(b + a sin t) -

π/

(b + a sin t) cos t dt

−π/

b

t+

b

sin(t) −

a

cos t

π/ −π/

= π a b

5" & S T%D!3 ^ c! & %! B {QB ^ TV+ t& d < b 5" & & .C a

< d % A74 && 4 VB V 24 =S

θ a a −x −y = √ − cos θ = a sin

dl = a dθ/ && θ + dθ θ @5;/ ,X :X L =%! T%! dA = zdl && p % T^&& =%! + =dA = (a /) sin(θ/) dθ 5 -

π

π

dθ = a sin

π θ = −a cos = a

A =

-

m B & VB %! LO * N $ w %+& VB S -VD& r 5" & ^+L @ Y;! L h -%! N$ N T" < +h< !d & <3 @+ % w0 =+&+&

2 N

@C/ 5" Tδ QG$ & ^G.P @ ! + i: < \3 , w %&E <L %C! & h < =V+ D!3 ! DB * =V+P ! 3 TV V+! ! U;N V+ i: %! ^"< =V / P x 3 & C/ < P y 3 ! p +; OrBh @ ! < L T%Q ^ ="

@L& & ! < L ! 5;/ ^ B =V+ D!3 %! " 4 [x; x + dx] x && < 3 d @74: < %! dm = πxhδdx && r

! !

x / + y / = a /

@D!3 w + V @D!3 w[Q 0=F 8" DB *

θ dθ

00

HP 1 2,

W "#:*8 ,

[Q`9=F 8" &x %! x = a d & 3 mH ^+& <3 % V+P 7 ^ mH =w" B V+ D!3 x = a x = L y = x < y = a

√ x ax − dx a a √ x/ x a a − = / a -

S

=

=

a

+&+& x + y = a @ ! L ./ d % w9 xOy @34 & < %! \& y = z @34 s! =%! <3

5" & V+ Y;! & < : <+ %&7; w6 Vh/ 2j& < %! MD; d Y;! & d ;/ r =%! <Xl d

2 N

s ^+& <3 % =+ D!3 x − x V+P & .! < s & Tp ^ & =

y =

.! < x + y

√ ax

x+y = y = x − x

=

⇒ x = x ⇒ x =

x=

@+ 8" Tx = .! c@ 8 & B & mK! 7 mH =w" B `9=F 8" &x V+ V+! p L y = −x < y = x − x ^+& % @D!3 L %DC x = x = S

=

- (x − x ) − (−x) dx

=

x

−

x

d %+. & L @p3Q P %C! w> @;& ^++5 %&7; =%! {! < S m %+. N (t = ) L ǱD 4 =" [j d N ,! T = %"]S L mH

=

@B t d R< 2 %! " 75 & & wF && S ! − ~

− ~ × t + ~ × − t & d S ' S & + ( =%! $ B 4 & d ! < B 4 =%! LO 2 _+ h

& 4 L s! " wI =+ D!3 " H

& : ' ! ' : & 4 w1 ! W" : ^ < V+"& " L+ <+ S7+ ' d : & * _+ $ T"& " ,X -%! L+

GN 0

0 ,e w

,e w[Q 9=F 8"

1*

+ i: x I FG & )&/- G T3 $ " ! & <
^+& <3 % T P " !
S=

b

f (x) − g(x) dx

a

)

y=x ,

x + y = ,

%! && [x; x + dx] @L& & ! +; % L =%! w=" B `>=F 8" &x =dS = f (x) − g(x) dx &

)

y = /(x + ),

x = y,

< ax =

)

x +y =

)

(x/) + (y/)/ = ,

)

y = x,

) y = − x / ,

,

y = x /,

x =

y = x + sin x,

(y −

y = ,

2

.! < ^+& % " ! =
),

≤ x ≤ π. 0>

y

ay = x ax = y

⇒ ax =

x a

⇒ x = a ⇒ x = a

W "#:*8 ,

HP 1 2,

√

√

V+ D!3 y = y = − L S

= −

= −

−y

√ √

-

=

#

√

-

−y

√

−

$ dx

y

y

dy −

y

)

√

−

)

√

y + arcsin √

−y

√ √

=

+ arcsin

=

√ + arcsin

D!3 " @ & +3 L ' P & <3 % +

−

−

√

−

√

x / + y / = a / , x / y / ) + = ,

√

√

)

x +y =a ,

y = x(x − ) ,

)

y = (x − )(x − ) ,

)

x + y = x + y .

+ i: y I FG )&/- & G T3 $ " ! =g(y) ≤ f (y) < !+H [a; b] @L& & x = g(y) < x = f (y) s Tg U& Tf U& . ^+& <3 % 24 ^ & %! && y = b < y = a -

b

S=

f (y) − g(y) dy

a

3 < & <3 % T P ! " ! + D!3 " √ √ x+ y = ,

) x+y = ,

y

)

x=y ,

x=

)

x = − y /,

x = y /,

)

x + y = ,

y = x,

+ ,

=" B [Q`6=F 8" &

2

< P y 3 ^+& <3 % " ! =+ D!3 x = y ( − y) 3 L %! ! @B L wy { &x x U& " B = : ^&& =w-$x %! c. y = < ]S y = y = y = L x = < x = y ( −y) ^+& % %! V<+& %!&

)

x − y = , x + y = , a+ a −y − a − y , y = , ) x = a ln y

- S

=

=

D!3 " +3 L ' P & <3 % +

)

a x = y (a − y ),

)

x = ( − y ) ,

AB8 )&

) (x

&

G

-

β

α

−

=

& 4A T3 $ " " !

(r (θ)) − (r (θ))

y

(a − y ).

(r (θ)) − (r (θ)) dθ

dθ dS = π π(r (θ)) − π(r (θ))

U& < . ^+& % w[Q 6=F 8" ,e w s! x

dθ

i< U& < . ^+& L " B %./ % L && TP N && Tθ + dθ < θ +p P5" s! dS =

y

!+H U& < r = r (θ) < r = r (θ) + i: 2;< B F=F 8" &x =r (θ) ≤ r (θ) T P [α; β] @L& & Tr = r (θ) U& . & <3 % 24 ^ w=" & %! && θ = β s < θ = α s Tr = r (θ) U& . S=

{y ( − y) − }dy

− y ) + y = ,

) x = y

)&/-

2 N

@ L ./ % =+ D!3 " & y = x V+P & .! < & p ^ & =

& %!

+y

x +y = y = x

=

⇒

y

+y =

⇒ y = − ±

√

=" B `6=F 8" & =y = ± y = mH x = y / < x = − y ^+& % %! : T5 09

HP 1 2,

[−π; π]

W "#:*8 ,

@L& %! 5 ^ &

⎡

sin(θ) ≥ ⎢ ⎢ sin θ + cos θ ≥ ⎣ sin(θ) ≤ sin θ + cos θ ≤ ⎡

−π

≤θ≤ ⎢ −π ⇒⎣ π π ⇒ ≤θ≤ ≤θ≤

D;/ U& < . ^+& % w[Q F=F 8" 1=I=F L %./ w

π

≥ θ ≥ t $ < ≥ r(θ) G d Tπ/ ≤ θ ≤ π/ S T 4 && r() < r(π/)
a sin(θ) (sin θ + cos θ)( − sin(θ))

= =

a

π/

-

π/

a sin(θ) (sin θ + cos θ)( − sin(θ)) cos θ sin θ (cos θ + sin θ)

2

3 < ^+& <3 @+ % " ! =V

V+ D!3 θ = π/ θ = L S

⎡ ⎧ ⎨ kπ ≤ θ ≤ (k + )π ⎢ ⎢ ⎩ lπ − π ≤ θ ≤ lπ + π ⎢ ⎧ ⇔⎢ ⎢ ⎨ (k − )π ≤ θ ≤ kπ ⎢ ⎣ π π ⎩ lπ + ≤ θ ≤ lπ + π

& %De " V

r(θ)

=

dθ

S

dθ

=

= =

-

=

z dz ( +z ) −a = a + z

a

⇒ cos(θ) =

< θ = π/ qθ = π/ + kπ/ θ = π/ + kπ 5 ^&& = / [−π/; π/] @L& θ = −π/

V u = tan θ i: & S

r=a r = a cos(θ)

=

=

- π/

S = (a cos(θ)) − (a) −π/

a π/ cos (θ) − dθ −π/

a π/ + cos(θ) − dθ −π/

π/

a θ − sin(θ) −π/

a

x + y =

dθ

√

π −

2 N

axy 2 & & <3 % =+ D!3 V& D;/ @34 & 3 & =

r cos θ + r sin θ = a(r cos θ)(r sin θ)

+

r (cos θ + sin θ) cos θ − sin θ cos θ + sin θ =

& & <3 % w[Q I=F 8" & 3 ' <3 % w

=

ar

r(θ) =

" !

06

sin(θ)

a sin(θ) (sin θ + cos θ)( − sin(θ))

^&& <

W "#:*8 , /

(y/b)

HP 1 2, /

= sin t < (x/a)

_+ < r

= cos t i: mH

5 = ≤ t ≤ π <

C : x = a cos t , y = a sin t ,

S

=

π

(a cos t)(a cos t sin t)

−(a sin t)(−a sin t cos t) dt π

a cos t sin t dt - π a = sin (t) dt = πa 3 L 8+! ' & <3 % D!3 P x 3 < y = a( − cos t) Tx = a(t − sin t) =+

ǱL& 3 5 T" 4 y N t = kπ ǱL& = 3 L 8+! ' mH = U;/ P x 3 t = kπ ,: ^+Q< L ! & ^&& =%! ≤ t ≤ π 5 & V )=I=F =

= −

S

π

= −a

π

+ cos(t)

− cos t +

t − sin t − sin(t)

= = =

- t

(x + y ) = a xy,

)

x + y = a (x + y ).

-

-

θ =

π

r − r < θ

t − t +

; & <3 % w ) =+ D!3

C : x = x(t) , y = y(t) ; α ≤ t ≤ β

S

dt

=

x(t)y (t) dt

-

β

= −

= −πa

α

=

β

y(t)x (t) dt x(t)y (t) − y(t)x (t) dt

α

@+A/ L ! & < < 7B L 1 j: ,: ^ 2DE B C 3 & ;N % %B & =S n; ^S `I=Fx " P −S B T"& m8C& %B S =" =w" B +h< !d & <3 %

2

" !

C : x / + y / = a /

) dt

β

= α

t + t t

=

3 & <3 % wJ

8" & & H 3 ' C + i:

t( − t)(t − t ) dt

3 L & ' % w1 =+

+θ =

-

t t ( − t) ( − t) + dt

t ( −

= a cos(θ)

( ) J 25A5 ^ G & 4A T3 $ # " !

2 N

=

)

& %! && C & <3 % T24 ^ =%!

& +Q & %fC =" πa && B T DQ

=%! 3 & ;N % %B & m8C < x = t( − t)/ 3 & <3 % =+ D!3 ≤ t ≤ y = t ( − t)/ V )=I=F ,: ^+< L ! & = S

x + y = ax y,

( − cos t) dt

)

=+ D!3 r

π

= −a

x / + y / = a / ,

a( − cos t) dt a( − cos t)

-

= −a

)

D!3 r

2 N

-

3 & <3 % w0

" @ & +3 L ' P & <3 @+ % + D!3

$

-

θ

3 < & <3 @N7 d % w> r = a cos θ < r = a(cos θ + sin θ) =+ D!3 T & (a/, ) @;N

#

3 < & <3 % w =+ D!3 r = a cos θ

=+ D!3 r = a cos

≤ t ≤ π

^+! L ! & + =" B [Q`1=F 8" & V )=I=F ,: -

= a( − cos θ)

p ^ & V+ H

0F

(x/a)/

C

=+ D!3 3 & & = V+

+ (y/a)/ =

]D " 1 I,

& y

= x/

U& . c/ ,X

W "#:*8 ,

2

!

=+ D!3 [; ] @L& V =1=F '. & =

- =

/ x

+

-

dx

x dx

/ × + x

√ −

= =

2 N

3 s! y = x / − . L !/ ,X =+ D!3 \& P x √ ^&& =%! x = ± x = 5 & y = " = & %! && p ,X

√

= −

)

x = a − b sin t,

y = a − b cos t

)

x = t − t ,

y = t − t

)

x = a cos t,

y = b sin t

)

x=t ,

y = t( − t )/

)

x = a cos t,

y=

)

x = a( cos t − cos(t)) ,

)

x=

a sin t

+ sin t y = a( sin t − sin(t))

c c cos t , y = − sin t , c = a − b a b

D!3 " @ & +3 L ' P & <3 % +

)

y = t − t

) x=t − ,

+

=

-

P = De C c < b Ta + i: " ! + D!3 " +3 & <3 % T

x + y = ax y

) x

/

+ y / = a /

+ x dx

√

√

= + x + ln x + +x √ − √ √ √ = + ln +

x

" !/ L ' P ,X ) y = ln x, a =

)

√

,

!

√

b=

[A 0

S x I FG ^ )&/- P/ /M ; A ! . c/ ,X T"& [a; b] @L& & ]HM 5& y = y(x) & %! && U& ^ -

+ D!3

(x+ dx, y(x+ dx))

y = ln |coth (x/)| , a = , b = c ) y = c ln , a = , < b < c. c −x

)

y=

+ y (x) dx

a

)

< (x, y(x)) N & <3 c/ ,X L &&

d = =

− ln x + x − ,

x x −

b

=

y = arcsin(e−x ), a = , b =

(dx) + (dy) dy + dx dx

=" B 1=F 8" & =%!

a= , b=a+

S y I FG ^ )&/- P/ /M ; A ! . c/ ,X T"& [a; b] @L& & ]HM 5& x = x(y) & %! && U& ^ =

b

H*

+ x (y) dy

=

a

0I

=I=F ,e L %./ w[Q 1=F 8" c/ ,X D!3 w

W "#:*8 ,

]D " 1 I,

^&& < y = r sin θ < x = r cos θ L

d

=

+ #

dy dx

dx =

dx + dy

-

(r (θ) cos θ − r(θ) sin θ)

=

=

+(r (θ) sin θ + r(θ) cos θ)

=

dθ

2

= =

= =

θ

- π a θ θ + ÷ cosh dθ a tanh - π θ a + tanh dθ

a

a

-

π

-

π

a (π + tanh(π)) /

L r(θ)

= =

= =

/

√

=

√

=

z dz z − / z − z + ln z+ √

) dy

−

=

2 N

+

− sin y cos y

dy

a

dy cos y a + tan y ln cos y a π ln tan +

=

2 N

a

=

dθ

U& . c/ ,X T P ! ! + D!3 b a L x = x(y) ) x = y / , a = , b =

−

√

dy

U& . c/ ,X T ≤ a < π/ + i: =+ D!3 y = a y = L x = ln(cos y) V 9=1=F '. & =

=

dθ θ θ (z − ) + (z − )

+ y dy

−

-

/θ

(y + )/

× × ( + y)/

=

U& . c/ ,X =+ D!3 θ = V z = + ( /θ) i: & =

=

+

/

-

−

dθ

cosh (θ/) π θ a θ − tanh

=

θ

-

+ tanh

−

=

+ (y +

-

@;& & r(θ) U& . c/ ,X ! θ =+ D!3 θ = π θ = L a tanh V I=1=F '. & =

=

+

−

$/

=

2

U& . c/ ,X ! =+ D!3 y = y = − L V 9=1=F '. & =

x = (y+ )/

) −zdz (z − )/

)

x= x=

y

, a = , b =

c − y y y −

c

− ln y + y − , a= , b=c+

√

√ √

)

y

− ln y, a = , b = e b + b − y ) x = b ln − b − y , ( < a) y

/

+ ln − ln( − √)

x=

2;< FG ^ )&/- P/ /M ; A " !

a

L r U& . c/ ,X T P

!

+ D!3 b

G d T"& [α; β] @L& θ L ]HM 5& r = r(θ) S & %! && r = r(θ) D;/ U& . c/ ,X

) r(θ) = aθ, a = , b = π

)

-

β

=

r(θ) = a( + cos θ), a = , b = π

α

01

r(θ) + r (θ) dθ

7$ ?P= HP $ U 1 J,

W "#:*8 ,

) C : x = a( cos t − cos(t)) , y = a( sin t − sin(t)) ; t

≤ t ≤ π,

C : x=t , y=

)

C : x = (t − ) sin t + t cos t ,

(t − ) ; −

√

)

y = ( − t ) cos t + t sin t ;

)

≤t≤π

)

π π , a = , b = r(θ) = a sec θ −

C : x = sinh t , y = cosh t ;

)

C : x=

≤t≤a

c c cos t , y = sin t ; a b

-

b

=

c =a −b

2+

x (t) + y (t) dt

a

6#
+h< !d c/ ,X

I*

I 2G C) & I 78 3 ) & U3 ; A !

!+H 5& y = y(x) + i: 6 y ) /A /3 x ^ =y(x) ≥ x ∈ (a; b) P ǱL& < %! [a; b] @L& & . & <3 @+ < L 4 < VB V 24 3 , x = b s < x = a s TP x 3 Ty = y(x) U& & %! && P x V =π

, a= , b=

< x = x(t) ]H+ U& '. & C 3 t $ ,X 24 ^ T" +& T%! a ≤ t ≤ b y = y(t) & %! && C c/

≤t≤a

-

r

( ) J 25A5 ^ P/ /M ; A # !

)

≤ t ≤ π ,

θ(r) =

& d N R 5" & @QQ ' + i: %!& " @+ % & QQ V %D = / h =
√

≤t≤

C : x = a(sinh t − t) , y = a(cosh t − ) ;

r+

)

2

!

C : x / + y / = a /

=V+ D!3 3 L %./ d ,X %! : p ^ & V =I=F .P ^&& =V+ D!3 %! ,< U& w%! C $ ' C x C : x = a cos t , y = a sin t ;

≤ t ≤ π/ ^&&

b

y (x) dx

a

=

% & %! && [x; x + dx] @L& & g @ ! V L & %! && dx y(x) 5" & @ dv = πy (x) × dx

=" B J=F 8" &

=

=

-

(−a sin t cos t) + (a cos t sin t) dt

π/

a

-

π/

y = t − t Tx =

sin t cos tdt = a sin t

√

π/

= a

2

t 24 & C 3 c/ ,X =+ D!3 %! " H − ≤ t ≤ & V )=1=F & B & =

=

- √ −

- = =

t

−

+ ( − t ) dt

( + t ) dt

t + t

−

=

L +3 L ' P c/ ,X 0J

!

+ D!3

7$ ?P= HP $ U 1 J,

W "#:*8 ,

DN % &

@L& & g @ ! L "& d Q ,X %! πy(x)

< V D B % @D!3 J=F 8"

[x; x + dx]

=

y=

V ! & %! && P x 3 , ≤ x ≤

+ (y (x)) dx V

=

2

S

= = = =

- √ √ π x( − x) + √ − x dx x - √ π x( − x) x − x + dx x - π ( − x)x − x + dx π ( x − x + ) x − x +

arcsinh

+ =

π

√

√

(x −

=

π

2 N

V

=

R

π −R

R −x dx

π R x−

=

x

R

= −R

πR

& <3 @+ < L 4 V ! D!3 P x = b < x = a s TP x 3 Ty . +

) y = x − x , a = , b = ,

2 N

&& R 5" & @ P Y;! % +P =%! πR U& . < L Y;! ^ . i: = & B & =d %!& P x 3 , y = R − x ^&& T%! [−R; R] U& ^ @ 8 S

=+ D!3 R 5" & @ V =−R ≤ x ≤ R < y = r − x V+ i: = _ < R 5" & @.+ L %! 2DC y . T24 ^ L 2DC V ^&& =P x 3 O& U/< < ǱD P x 3 , .+ ^ L % < L 4 V & %! && T"&

+ ln( + ) +

/ 0 0 −x 1 = π R −x + dx −R R −x - R = π R dx = πR -

π R x − x/

=

) √

- √ π x dx

3 < L 4 Y;! % ! =+ D!3 P x 3 , y = x( − x) U/< T%! N P x 3 & %D 3 = d %!& y / 4 && L =y = ± x( − x)

@L& & y = x( − x) U& mH =x = x = V+& V+ mH ≤ x ≤ $ =V+S p [; ] √ + Ty = x( − x)

2

√ x .! < L 4

! !

+ L 3 < L 3;! % %&7; w x = @74: P x 3 , y = a cosh (x/a) =d %!& x = a

y = (x/)

, a = , b = ,

)

y = sin x, a = , b = π,

)

y = cosh (x/c) , a = −c, b = c.

3 < & <3 @+ < L 4 V w6 D!3 P x 3 , y = x/ < x + y = =+

3 < & <3 @+ < L 4 V wF =+ D!3 P x 3 , y = /π < y = sin x

R

−R

/

)

I 78 3 ) & U3 N< T3 $ ; A !

y = y(x) + i:

6

x ) /A /3 x I 2G C)

&

ǱL& < %! (a; b) @L& & ]HM < [a; b] @L& & !+H 5& Y;! V % 24 ^ = ≤ y(x) x ∈ (a; b) P x = a L x y = y(x) U& . < L 4 < VB & %! && P x 3 , w%! " V+! x = b S = π

>)

-

b

y(x)

a

+ y (x) dx

- . / 0*1 V,

=

πab

W "#:*8 ,

V+ < L 4 8" < Y;! % %&7; w0 =P x 3 , y = sin x !+! 3 L |

−b +a b −a b −a b + arcsin b −a b −a b

3 < L 4 Y;! % %&7; w> =P x 3 , [; π/] @74: y = tan x +h< !d < L 4 Y;! % %&7; w9 =P x 3 , x / + y / = a /

!

3 & <3 @N; < L 4 < VB V w =+&+& P y 3 , x = √y < x = y +h78+! < L 4 y = L < P y 3 ,

3 < L 4 Y;! % %&7; w6 =x = e x = L P x 3 , y = x /−(ln x)/

< VB V w0 x = y /(a − y) =+ D!3 y = a

+, - ./(0

)&/- C) & I 78 3 T3 $

U& . S 6 y ) /A /3 y I 2G Y;! % TV+P < P y 3 , a ≤ y ≤ b

& && V 4 < VB V < S 4

x = x(y)

U*

S = π

K+

Int(exp(−x) ∗ cos(x), x = 0..Pi) −−−−→ 0

π

e

x(y) + (x (y)) dy,

b

(x(y)) dy a

2

@7+!& y = ax .! L ./ ! TV+P < P y 3 , %! " B x = a s & %! && 4 VB V V

=

a

π − a

y

a

y π × a

=

+1

dy

a

= − a

A+& < L 4 Y;! % % & %! && P y 3 , x /a P y 3 , [−b; b] @L& & x = a

e−x cos x dx

K+

value(Int(exp(−x) ∗ cos(x), x = 0..Pi)) −−−−→ 1 −π e +1 2

7 24

b

a

V =π

! 7 24 E- ; A # ! f (x) d T%! int(f(x),x=a..b) ,G @D!3 int & S =%! +GQG +a x < ,G U& " ^. 8" & ,G T" ! Int L ! L %! : d @D!3 & =" P( D!3 < . & =V+ ! value K+ 1 −π 2

-

-

%3 W(& & TK+ _: L ! 2N @P & =" 5B ' j: L ^+.P

int(exp(−x) ∗ cos(x), x = 0..Pi) −−−−→

U3 " !

S

=

= E- )& b b # !

8" & ,G +a ++a !

=

student[changevar](R(x,u),I(x),u)

x +a { & %! QG I(x) d T%! %! ;& R(x, u) TV+P d +a ++a V+P %! B +a u < V . +& x { & u d

'. & T . & =V+& d { & ,G &

=

π

-

b

a −b

-

2 N

+ y /b − y /b

y dy b

− y dy

−a b −a

−y

b y b + b −a arcsin b (b − a )

−b

>

=

3

/ # $ 0 y 0 −ay/b 1 − + dy b − y /b

b −a − b −b / 0 - b0 πab 1 −b b −a b ⎡ / 0 πab ⎢ y 0 1 ⎣ b −a b

πa

b

πa

W "#:*8 ,

- . / 0*1 V,

-

!

π

i: & Ǳ_B & Ǳ_B \< L x/ cos x dx ,G =" ! dv = cos x dx < u = x/

student[changevar](u^2=x^2+1,Int(x^3 √ K+ - 2 2 2 *(x^2+1)^(1/2),x=-1..1),u) −−−−→ √ −(u −1)u du − 2

E- 2; )4 ; A # !

+p T. 7+73 MX & QG + i: - π evalf ! '. & 24 ^ T sin x/x dx . & =. D!3 DN X & d C N K+

u

=V+! 24

evalf(int(exp(−x ∧ 3), x = 0..1)) −−−−→

< . D!3 L

−x

dx

cd

u = u(x) -

!

# !

http://webpages.iust.ac.ir/m_nadjafikhah/r1.html

=%! "
-

√

−

√

- −

x (x + ) dx

,G

−(u − )u du @+ & < .

E- )& ǱHK G ǱHK ? ) # !

student[intpart](I(x),u(x))

,G DN N =%! + ~ e

+a ++a L

8" & ,G Ǳ_B & Ǳ_B ! 7

0.807511182

-

= x +

" i: I(x) ,G d '. & T%! u dv = uv − v du Ǳ_B & Ǳ_B @C/ L mK! < ! '. & . & ="

student[intpart](Int(x^(3/2)*cos(x), K+ 3 π√ ( x = 0..Pi), x 3/2)) −−−−→ − x sin x dx 2 0

>0

% " #$ "$ -

a

f (x) dx

$" ! "!

b→−∞

! h" $" &c(! "!

y = f (x) U& < I = [a; b) + i:

"

%! ]KQG I & f V+hS 4 =S [5 I & 0= =F 5 & f U& Ta < c < b (Q c P ǱL&

B c→b− lim I

-

c

f (x) dx

f $" ! "! a

^+5 ,G T!! i: < ,.C & V" j: +GQG @ V i: 8 ,< =V 3& i: 8 d < < w[a; b] 8" & & @L& 'x %!

j: ^ =%! [a; b] & f ,G U& TV

! QG @ L < V& p Pi: wC +l T5x ! QG + & wC T5x G >= =F @ ! / T+Q ^+.P & =V+. S =%+ ,D/ & ! QG T3& +& %Q! & 8+& @ d T8 =V+ V+N 7 < P %! ^"< = 8+& U& d G < < ^ L P . . & G ! ,G =. _

a

< %! I = (a; b] + i: " 4 =S [5 I & y = f (x) @" U& a < (Q c P ǱL& %! ]KQG I & f V+hS

-

b

f (x) dx . & < + I

-

b

f (x) dx

f $" ! "! ] c

=V+P D/ [5 _+ < [5 ^ " :5 ! QG =V+ T

a

. +∞

F h" $" &c(! "! + ) ,G N 2

"

dx/(x

=+ D!3 B< 24 i< U& G d T < b S %! ^"< T & =

b→+∞

b

dx = lim x + b→+∞

arctan

x b

5!4

1

L P 5 L C. 2j& & QG P T%" L n< ,G $ T5x =V+ w

$kl" ! "!

j

&

< I = [a; +∞) + i: " I & f V+hS 4 =" [5 I & %! 5& 5 & f U& Ta < b (Q b P ǱL& %! y = f (x)

-

b

lim

f (x) dx

$" ! "!

b→+∞

f

\](! "!

a

a

< I = (−∞; a] + i: " I & f V+hS 4 =S [5 I & "& 5& 5 & f U& Tb < a (Q b P ǱL& %! ]KQG y = f (x)

>>

/#18 "#:*8 2

6 # 5 2

-

− lim

=

ue

( )

−

=

−u

#

d→+∞

=

du

5d 4 lim − ue−u −

d→+∞

4 5d d + e−u d e d+ d→+∞ lim − ed d d→+∞ lim d − e

d→+∞ lim

= = = ()

d→∞ lim

=

b arctan − arctan b→+∞ b π = lim arctan −

d

-

$

d

e

−u

b→+∞

du

- ln x √ dx = − x

B< 24

(e

×

−

[a; ]

&

ex

- lim

a→−∞

+∞

e− √

- − √x - ∞ − √x e e √ √ dx = dx + dx x x x

ex dx

=

b→+

− lim

=

b→+

√

,G T
x

e− x √ dx x

=

lim

b→+∞

()

=

= =

lim

b→+∞

lim

b→+∞

B< 24

-

+∞

√

e− x √ dx = x

−e

−b

e

lim

c→−

c

=

−

-

b

√

x −x

x dx/

dx

B< 24

" i: w x _+

−

e

+

e

()

=

2 N

−x

-

( −u ) (−u)du u d→− - lim ( − u ) du d→−

=

√

x

d

lim

=

=

-

e

√

- -

e−u du

b→+∞

−

ex dx =

^&& =u =

4 −u 5b −e

a→−∞

.

e

dx

x

− lim ea =

=

√ − x

√

a→−∞

,G N =+ D!3 = i< U& G d T < c < S %! ^"<

− lim e−b − e− =

√ x %!

+ =u =

b

a→−∞

−∞

%! " i: w x V < -

√

∞

e

−

lim [ex ]a lim e − ea

=

-

b

b→+

2 N

=+

S %! ^"< =

<

a

4 5 lim −e−u b

=

N

^&&

- − √x e √ lim dx x a→+ a - lim e−u du

( )

π dx = x + . x e dx ,G −∞

=

V T,< ,G =

+∞

2+ N

√ x

- − √x e √ dx x

^&&

=

√ / x) dx ,G N

U& G d Ta

=+ D!3 G < 8 T 8 B < ,G ^ = V+ V+N %./ < & ] ,G T^&& =+∞ -

π

D!3 B< 24

√

√ − x

π

-

L w0x T x = e−u %! " i: w x T8 Y+ , +HP @C/ L w>x < %! " ! Ǳ_B & Ǳ_B \< ^&& =%! " :S &

.∞

π

=

− = −

ed

lim

√

d

( − u ) du (u − u + )du

u

−

u

+u

− x %! " i: dx =

−x . √ (ln x/ x) dx

w x

2 N

,G N =+ D!3 ,G U& G d T < c < S %! ^"< = √

=

lim

>9

u

du

6 # 5 2

/#18 "#:*8 2

2

N ǱL& V+ & V+P " " .∞ p -%! G.P dx/x ! ,G < p N+N C L V T ! ,G [5 & B & = -

∞

- b dx = lim x−p dx b→∞ xp

V+S p L %Q ! Tp N & & G.P /(p − ) C & ,G 24 ^ p > L T%!

2

-

∞

dx xp

= = =

lim

b→∞

x−p −p

b−p − b→∞ −p

∞

dx xp

= =

∞

=

−∞

√ - √ x - +∞ √ x ex e e dx = dx + dx x + ex + e + ex −∞ √ V u = ex +a ++a & T P

−∞

√ ex dx + ex

p−

=

2p = ;!

= =

b

lim [ln x]

b→∞

=

lim ln b = +∞

2

lim

b→∞

x−p

;!

-

−p

+∞

b

b−p − = lim b→∞ −p

√ ex dx + ex

= = = =

2 N

-

∞

-

= ()

=

= = -

∞

xp dx +x

- ∞ xp xp dx + dx +x +x - - ∞ u−p −du xp dx + +x + u u +∞ - ∞ −−p - ∞ x xp dx + dx +x +x J−(+p) + Jp p

x dx mH =Jp = < u = %! " i: w x

+x x p & %D Jp G d T ≤ p S =" !& Jp %! :

lim

b→+

π

[arctan u]b

lim (arctan

b→+

−=

− arctan b)

π

√

b

ex dx b→+∞ + ex - a u du lim a→+∞ +u u - a du lim a→+∞ +u lim

b a→+∞ lim [arctan u] a→+∞ lim (arctan b − arctan

π−

π

=

)

π

+

N+N C L N ǱL& V+ & V+P . -%! G.P ∞ xp dx/( + x) ! ,G Tp V+ B p ^ & = =

=

= +∞

- √ x e lim dx a→−∞ a + ex - u du lim b→+ b +u u - du lim b→+ b u +

-

=

+l < %! G.P " ,G G d T < p S + ="& S< T24 ^

Ip

= =

dx xp

+∞

-

L T%! S< ,G _+ 24 ^ < p < -

-

lim

b→∞

2# N

.∞ √ ( ex dx)/( + ex ) −∞

,G N =+ D!3 B< mH =%! 8 −∞ _+ < +∞ ,G ^ = V+

b

L T%! S< ,G 24 ^ -

;!

24

-

+∞

−∞

√

ex π π dx = + = π x +e

"& " B< ! ,G ! " =V+ G.P +l ! ,G =V+

!;!

!0

! ,G ' 8 ^ ^ : L +& N fk.C ="& V +& TS< %! G.P i< = . G.P ! QG L +& Ty: QG z $ :+H Pp '. & DQ = D!3 d L & QG S< < G.P @7 & T5& ^ ^&& =V+ + d M+/ N @D!3 & L < VLH ! >6

/#18 "#:*8 2

#:L <@8.a 2

=V+P / 5Q; @ W_: & G d T"& %De 5& y = f (x) S -

< Txq ≤ xp G d T

-

b

& f (x) dx 5 T& W_: ,G N T+GQG a & d +Q ^+.P & =%! 54 b L 5& C - ∞ f (x) dx ! ,G B< 5 N+/ +& & T%! a

& y = f (x) U& + i: T; FG C/I= " LO " T24 ^ =%! ]KQG < %De [a; ∞) @L& -

U& %! d

∞

f (x) dx a

-

Jq

≤

J

=

f (x) dx

-

∞

f (x) dx

=

a

=

N

lim

N →∞

f (x) dx #a-

lim

b

a

-

b

=

f (x) dx + lim

N →∞

a

≥

$

N

f (x) dx +

N →∞

-

-

T−

+

b

f (x) dx

<

∞

dx +x lim ln |b + | = +∞

b→∞

Jp

N

=

b

<

f (x) dx = I(b)

()

=

∞

=%! M = f (x) dx & O& L I(b) T^&& a > a P L & "& M V+ i: , V+ i: =I(b) ≤ M

= < =

∞

-

f (x) dx

mH T%! %De f $ T24 ^ a A ^&& < M ∈ A T V+ i: = f (x) dx = α V+ %&E a P L & =α ≤ N < α + ε %P N ∈ A B< b > a T a L & S 24 ^ +l TL =I(b) > α − β [5 vf α − ε ∈ A G d TI(b) ≤ α − ε b > a P b > a P L & T%! 54 I $ + =%! α P& T{+ ^ & < |I(b) − α| < ε α − β ≤ I(b) ≤ α − ε 2 =%! .

b→∞

-

b

A := N ∈ R ∀ b > a : I(b) < N -

b

lim

n+

a

-

x dx +x

S =%! S< Ip ]Q < %! S< Jp G d T ≤ p S T^&& mH = P S< Ip < J−p− ]Q < − −p ≥ G d Tp ≤ − %Q ^ L T" 5Q; − < p < %Q %! :

8 & B & mH =+S / (− ; ) @L& _+ − − p (− ; ) @L& − + ;− + 8" & P L& n+ n

b

-

xp dx = Jp +x

∞

=

T24 ^ ="& B f (x) dx V+ i: A7S" a < %! %De L& P & d ,G T%! %De 5& f $ b > a P L & T+

b

-

a

∞

xq dx +x

S< _+ ≤ p P Jp @.P G d T"& S< J S mH = P

G.P & : <

="& O& L

b

-

b

-

=

=

I(b) :=

-

< ≤ q ≤ p S L T%! 54 ^&&

≤x

n

V+ i: T "H 24 ^

xp dx +x

x−+ n+ dx +x - ∞ y −n (n + )y n dy n+ + y - ∞ dy (n + ) + yn -∞ dy (n + ) n y +∞ y −n (n + ) lim b→+∞ −n ∞

f (x) dx ≥

=

S $ =x

n+ n−

= y n+

%! " i: w x 8 Y+

p ∈ (− ; ) P ǱL& Jp ^&& T n + ≤ G d Tn ≤ n− Ip ]Q < "& G.P _+ J−(+p) %Q ^ T%! G.P %! G.P /< < /< Ip " %&E 5 =%! G.P =− < p <

!DM 9=7-`

1

< G.P '. %! +A/ Ld L p V+N +l < V+N 24 & ! QG S< >F

#:L <@8.a 2

/#18 "#:*8 2

.

T =0=I && mH ="& G.P a∞ g(x) dx V+ i: Tp P ^+t.P < b ≥ a P L & :X L =%! O& L ^&& =f (x) ≤ g(x) x ∈ [a; b] -

-

b

b

f (x) dx ≤

g(x) dx

a

-

^&& <

a

-

∞

≤

f (x) dx

a

="& G.P T+ < %! _+

2

2

@! ,G G.P

-

-

∞

dx +x

-

f (x) dx

xn dx − x

= +∞

=

=%! G.P

≤ u−

f (u) =

2 N

un

≤ =

un

u − u un

u/

+ u −

u − u

g(u) du = =

du u/

b − lim √ = b→∞ u lim

b→∞

I(b)

b

e−x dx

e−x dx .

∞

4 −x 5b −e e− − e−b ≤ e−

^&& < I(b) ≤ A +

x− dx x +x -

; ∞)

/e +

,G G.P

& f (x) =

x− x +x

2 N

$ =

b

=

5 =%! %De %+& && b → ∞ /< h(b)

=%! S< ,G ]Q < %+

S< _+

= g(u)

∞

dx +

U& < g(x) < f (x) + i: $ C/I= " ' ǱL& S & T"& [a; ∞) & ]KQG < 24 ^ =f (x) ≤ g(x) x ∈ [c; ∞) P < c ≥ a

b

-

x− dx mH x +x ^&& < x − > x − mH T ≤ x 8 d +Q & x +x x +x - b x− I(b) ≥ dx x

b ln x + = x = ln b + − = h(b) b

+ u −

(u − )n

-

=" 3& & I(b)

(u − )n du

u − u

−x dx e

T%! ]KQG < %De (

−du − ( − u ) u

≤ u

un

e−x dx =

.∞

=+ 3&

( − u )n

+∞

-

=

^&& < ≤ u T + T−u ≤ − ≤ −u + u −

^+t.P =

b

=+ 3& − x) +a ++a L ! & =

-

-

-

b→∞

/(

e

−x

w x < %! C ,G ' A = e−x dx

Tx ≤ x G d T ≤ x S %! " ! %+5/< ^ L :X L =e−x ≤ e−x ]Q < −x ≤ −x

b

,G G.P

e−x dx

A+

b→∞

xn dx − x

V u =

≤

lim

-

=

"

b

a

S< _+ " ! ,G V+S + T%! S< =%! ∈ N

=

-

dx x + lim ln |b + | = +∞

= =

Tn

I(b)

∞

∞

"

=+ 3& T%! ]KQG < %De [; ∞) & f (x) = e U& $ = .b I(b) = e−x dx %! : %De U& Ld && =V+ %&E

( )

-

2

,G G.P

−x

+

=V+ 3& dx/ +x + x ≤ ( + x) G d T ≤ x S = /( + x) i: & mH = +x ≤ +x 8 & B & < =0=I g(x) = / + x

^&& < < f (x) =

−x dx e

-

∞

g(x) dx

a

.∞

_+

-

-

a

∞

g(x) dx G d T"& S<

∞

f (x) dx a

G d T"& G.P

-

∞

a

-

∞

f (x) dx S

w[Q

=%! S w =%! G.P

g(x) dx a

m8C w[Qx T =0=I & B & %! ^"< A7S" ^ & =V+ 2DE wx %! : mH =%! wx +N >I

/#18 "#:*8 2

#:L <@8.a 2

lim

=

+ x

x→∞

_+ < -

-

∞

g(x) dx

=

∞

−/

x

=

-

lim

b→∞

=

lim

b→∞

b

dx −/

x

%&E ^

=+ 3& &

∞

f (x) =

dx xex + e−x

xex + e−x

lim

x→∞

b − √ = x

< /<

2 N

< g(x) = e−x i: & = x 8 & p < 6=0=I [ ; +∞)

=

lim

x→∞

xe xex + e−x

xe x x→∞ xe x + (x + )ex lim = x→∞ (x + )ex

=

lim

P

=

-

∞

g(x) dx

=

≤ =

∞

x

+∞

e−x dx

b→∞

-

g(x)

∞

! ,G G d T"& %+&

f (x) dx a

="& G.P

G d T"& G.P

G d T"& S< <

-

∞

-

∞

<

g(x) dx -

L

∞

f (x) dx

<

e

2 N

! ,G q < p N ǱL& -%! G.P q < < q = T < q %Q ! Tp ^ & = =V+S p

xp eqx dx

g(x) = < f (x) = e

S w -

∞

f (x) dx a

L = ∞ S - ∞

=%! S< _+

w| g(x) dx

a

G.P T+ = Lg(x) < f (x) < Lg(x) - ∞ - ∞ < %! L f (x) dx G.P 5 & g(x) dx a a - ∞ - ∞ g(x) dx S T

L

a

∞

V+ i: G d T < q S w[Q =p < 8 < ≤ p 24 ^ =xp eqx

-

∞

f (x) dx ="& S< _+ a $ k Tε = L & TL = " i: S , ≤ f (x)/g(x) < ε x ≥ k P L & B< - ∞ g(x) dx G.P T^&& =f (x) ≤ g(x) & -

∞

=" f (x) dx G.P a B< $ x ≥ k TL = ∞ " i: S , S< T^&& =f-(x) ≤ g(x) f (x)/g(x) ≥- ∞ ∞ 2 =" f (x) dx S< & g(x) dx -

a

∞

qx/

%! ^"< T G d T ≤ p S

G d T"& S<

f (x) dx a

a

∞

=

=%! G.P _+

a

^&& < %! S<

G.P _+ " ,G V+S + T"& G.P =%! -

%! G.P /<

g(x) dx a

a

a

e−x dx

4 5b lim −e−x =

- n x dx − x

(L − ε)g(x) < f (x) < (L + ε)g(x)

_+ < -

=%! G.P

g(u)du mH - ∞ f (u)du

L & T24 ^ =L = , ∞ V+ i: A7S" x > k P L & B< $k > a ' ε = L/ + =f (x)/g(x) − L < ε

x

f (x) g(x)

U& < g < f V+ i: (43 $ C/I= " 24 ^ ="& [a; ∞) & ]KQG < %De f (x) 4 [Q( < B L = x→∞ lim S w[Q

dx

,G G.P

∞

="& G.P n ∈ N P ǱL&

="& G.P p ,G V+S + -

-

T =0=I M&; T^&& < %! G.P

=

dx + x

,G G.P − /

[ ; +∞) & g(x) = x

f (x) x→∞ g(x)

f (x) lim = lim p qx/ = x→∞ g(x) x→∞ x e

lim

>1

=

2

a

! "

=+ 3& i: & = < f (x) = + x 8 & p < 6=0=I lim

x→∞

√ x x

+ x

#:L <@8.a 2

/#18 "#:*8 2

V+ i: 6=0=I @+A/ L wx %./ 24 ^ =g(x) = ≤ x→∞ lim

f (x) g(x)

=

lim

x→∞

=%! S<

-

∞

e

I :=

< f (t) -

=

/ ln t - ∞

∞

g(t) =

e

lim

x→∞

lim

∞

dx x/

2+ N

,G +P V t = ln x i: & =

dx = x ln(ln x)

∞

dt ln t √ =g(t) = / t

n< & T24 ^

f (t) lim t→∞ g(t)

=

P = =

√ t lim t→∞ ln t lim

x→∞

lim

t→∞

√

P

(−p)(−p − ) · · · (−p − k)x−p−k = q k x→∞ eqx/ lim

f (x)

= 24 P T5 i: & T^&& =x→∞ lim g(x) < 6=0=I [ ; ∞) & g(x) = xp eqx < f (x) = eqx/ 8 & p -

-

∞

f (x) dx

= =

V+ i: 6=0=I L w|x %./ , dt ln t

−px−p− P = ··· x→∞ q eqx/ lim

=

-

P

=

=

x/

g(x) dx =

f (x) x−p = lim qx/ x→∞ g(x) x→∞ e

sin x/x /x/

-

∞

dx x ln(ln x)

∞

x

lim

≤ -

x

sin x x→∞ x/

=

,G ]Q < %! G.P

< f (x) = sin

x/

B< k = [−p] + 5+DX C G d Tp < S T L ! & k L mH T^&& =−p − k < V+! ≤ p %Q & k T, +HP @C/

∞

eqx/ dx

lim

b→∞

eqx/ q

b

=∞

L w[Qx %./ && - V+S + T%! S< ="& S< _+

∞

xp eqx dx

,G T6=0=I

/< < /< T w x F= =I M&; G d Tq = S w - ∞ =p < − "& G.P xp eqx dx

< g(x) = i: & G d Tq < S w| x & +D" QO ! &x 8 & p < I=>=I [ ; ∞) & w" u [Q %./ t d

√ / t /t

f (x) = xp eqx

t = +∞

f (x) xp+ = lim −qx = x→∞ g(x) x→∞ e - ∞ - ∞ g(x) dx = x− dx = < + T%! G.P - ∞ ="& G.P _+ xp eqx dx ,G V+S lim

G.P & : < LO " 2W/' C/I= " " - ∞ $ k ≥ a Tε > P L & %! d f (x) dx .

=

β α

f (x) dx < ε

a

α, β ≥ k P L & B<

L & mH =%! I & G.P a f (x) dx V+ i: A7S" b ≥ k P L & B< $ k ≥ a Tε > P . b V α, β > k P L & T = a f (x) dx − I < ε/ .∞

β f (x) dx α

- α β f (x) dx − f (x) dx < a a α β f (x) dx + f (x) dx ≤ a a ε ε + =ε <

%P k ≥ a Tε > P L & V+ i: . T24 ^ = αβ f (x) dx < ε α, β > ε P L &

-

∞

xp eqx dx ! ,G T. mH /< < /< =yq < z yp < − < q = z %! G.P

=%! G.P -

∞

-

∞

sin x dx ,G +P x

^ " & =

- - ∞ sin x sin x sin x dx = dx + dx x x x

,G ^&& <

sin x x→ x lim

=

8 & B <

V+S + T B< ]Q < T%! C =V+P >J

2 N

-

∞

- sin x dx x

sin x dx G.P %! :

x

/#18 "#:*8 2

)

-

∞

$ #N #:L 2

x dx − ex

-

)

∞

(ln x)p dx.

# !K !DM

V+hS T"& G.P -

∞

∞

|f (x)| dx

S

"

PQ &!0

a

-

∞

< G.P f (x) dx S =%! f (x) dx a a =%! V+hS T"& M7; G.P +l

G d T"& M7; G.P i< ! ,G ' S

@P & =%! %! S ^ m8C ="& G.P =" B F=0=I ,e L w x %./ & d L .

i &!0

+ <

- k ∞ ∞ f (x) dx − f (x) dx = f (x) dx < ε k a a

-

-

a

=%! M7; G.P

b

|f (x)| dx .∞ a

f (x)g(x) dx

+

.

& G.P a∞ f (x) dx V+S 7G= C/I= " 24 ^ ="& [a; ∞) & 8 < 5& g < . =%! G.P a∞ f (x)g(x) dx P L & T9=0=F @+A/ L w9x %./ '. & A7S" B< $ ξ ' Tt , t > a -

-

t

t

f (x)g(x) dx = g(t )

-

ξ

f (x) dx + g(t ) t

)

t

f (x) dx ξ

P L & B< $ M T%! g $ . L & T%! G.P a∞ f (x) dx $ =|g(x)| ≤ M x ≥ a t , t ≥ k P L & B< k ≥ a Tε > P . t Vu Tk < t < ξ < t $ T = t f (x) dx < ε/M

t

a

f (x) dx

&& < B k→∞ lim

.k a

f (x) dx 5

)

) )

t

9)

∞

x dx x −x +

∞

-

dx

)

+ 3& L ! QG G.P

-

)

f (x)g(x) dx =

-

x + x +

-

dx √ (x − ) −x - +∞ - ∞ dx x ln x ) ) dx ( +x ) x +x +x - +∞ - ∞ arctan x ) dx ) e−ax cos(bx) dx ( + x )/ - +∞ - ∞ dx ) e−ax sin(bx) dx ) x - - dx

) ln x dx ) − −x - +∞ - +∞ dx dx ) ) x +x− (x + x+ ) −∞ - +∞ - π/ dx ) ln(sin x) dx ) x + - π/ - xn ) ln(cos x) dx ) dx −x - +∞ - +∞ dx dx ) ) n cosh x x(x + ) · · · (x + n)

a

2

+∞

)

& B< M C T%! g $ A7S" .∞ f (x) dx $ T
G d Tb ≥ a S T^&& < |f (x)g(x)| ≤ M |f (x)| b ≥ a |f (x)g(x)| dx ≤ M

.∞

N+N C a + i: L 2. " β < αT q Tp < Y+34 C m T%De 5+DX C n T%De =P(Q N+N C G.P 24 x L ! QG L ' P N + D!3 w&

.

b

=%!

2

M7; G.P a∞ f (x) dx + i: " ^ ="& [a; ∞) & < ]KQG 5& g < & . =%! M7; G.P _+ a∞ f (x)g(x) dx 24

-

P L & V β = k ^ "

G %&E & . α mH T%! (Q α ≥ k $ = k f (x) dx < ε

- b ∞ f (x) dx = lim f (x) dx < ε k b→∞ k

1

= & %De U& TV+ " + Ld & W(& ^ = + ^+$ U& L +& fk.C Q =VLH 7 Ld $ u -

α ≥ k

∞

-

∞

-

∞

xp

dx + xq

) )

arctan(ax) dx xn

)

dx p x lnq x

)

xα |x − |β dx

)

-

∞

-

∞

-

∞

x

dx x +

m

x dx + xn

ln( + x) dx xn

xm arctan x dx xn +

-

∞

-

ln x dx −x

$ #N #:L 2

-

∞

<

sin x dx x

F (x) =

f (x) dx =

g (x) =

− x

lim g(x) = lim

x→∞

x→∞

x

(k+)π kπ

<

∀x>

sin x x dx

-

∞

-

sin x dx V+S x

≥

lim

π

k=

3&

π

-

n

lim - ∞

π

k

+x

F (x) =

x

=

f (x) dx

A7S"

a

ξ

f (x) dx +

a

.

f (x) dx

t

f (x) dx −

a

≤

-

ξ

f (x) dx

t

a

≤ M + M = M

t Tx→∞ lim g(x) = $ = ξ f (x) dx < M & 24 & B< $ k > a T (Q ε > P L & ^&& + =|g(x)| < ε/M x ≥ k P L &

dx x

t

f (x)g(x) dx =

=

g(t )

≤

|g(t )|

-

ξ

t

f (x) dx + g(t )

t

dx x

= =

T^&&

f (x) dx + |g(t )|

t

f (x) dx

ξ

ε ε × M + × M M M ε ε + =ε

=%! . P& <

2

T%! !+H [a; ∞) & f U& + i:

2 N

f (x) dx

ξ

ξ

t

cos x dx ! ,G G.P +x

& < 9=>=I g(x) =

t

t

dx = +∞ x - ∞ sin x S< x dx

=%!

< t < ξ < t B< $ ξ '

t

(k + )π - k+

T <

a

≥ a P L & T9=0=F @+A/ L w9x %./ &&

kπ

n→∞

=%! ∞

k= n )

n→∞

=

-

k= n )

:Q )& C/I= "

x

=

∞

lim

=

-

f (t) dt < lim g(x) = x→∞ - ∞ =%! G.P f (x)g(x) dx G d

Tt , t

(k + )π

n→∞

≥

f (x) dx

t

f (x) dx

t

ξ

t

a

sin x x dx π - (n+)π sin x lim x dx n→∞ π n - (k+)π ) sin x lim x dx n→∞

-

=

f (x) dx + M

f (x) dx

M ε = ε M

& F (x)

(k+)π

V

=

t

t

t

=%! . P& T" Ld && <

[a; ∞)

=

sin x x dx ≥

f (x) dx + |g(t )|

8 [a; ∞) & g U& + i:

+

| sin x| dx (k + )π kπ

(k+)π ± cos x (k + )π kπ

≥

∞

ξ

ξ

2

=

=

-

=

M

f (x) dx

t

f (x) dx + g(t )

t

& B & T
≤

M

-

ξ

t

∈ [−; ]

x

≤

|g(t )|

sin x dx

<

=

g(t )

t

x

-

= cos x −

! "

-%! < G.P " i: 9=>=I S L T7& = +Q& G d Tg(x) =

x

2

! ,G d

f (x) = sin x -

/#18 "#:*8 2

"

-

x

& g =%! [a; ∞) & F (x) = f (t) dt a 24 ^ =x→∞ lim g(x) = < g ≤ T]HM [a; ∞)

=+

i: = 8 & B

< f (x) = cos x

=%! G.P

-

∞

f (x)g(x) dx a

& ^. C & < T"& &d Ld L + A7S" 2 = @C

cos x dx

9

/#18 "#:*8 2

#*7; 4 *P4$ /#18 <@Q#:*8 2

!(6E 3 (N3# .!07 9=O!D(7

-

1

=

cfHO \< +p T ! ,G P& L +& & 2O5 : QG < Q.5 + 2O5

! QG { & 5& %! LO Th_B 2N & UB V+ & & Zj(& =V+ 5Q; " [5 ^ =V+ 3& d ]KQG < ]HM TG !+H =" VP: V ^ & LO _& W(&

-

∞

f (t, x) dt a

=%! G.P

x→∞

-

∞

< %! +l S ⊆ R + i: " 5& f (t, x) + i: =(a; b] × S = {(t, x) | a < t, x ∈ S} ,G x ∈ S P ǱL& < S [5 (a; b]×S & %! . V+hS 4 ="& G.P F (x) = ab f (t, x) dt < %! G.P F (x) & 88 ;&

-

b

f (t, x)dt a

∀ ∃c ∈ (a; b) ∀c ∈ (a; c ) ∀x ∈ S : - b f (t, x)dt < F (x) − a -

V+ %Q ^ b

f (t, x)dt = F (x) a

cos x dx ! +x

-

-

=%! 88 G.P S &

a

2 N

sin(x ) dx =

f (t) = sin t

-

L &

∞

sin t √ dt t

-

∞

∞

,

g(t) = √

t

sin(x ) dx V+S + 9=>=I @+A/ L

-$ =%+ M7; G.P ,G ^

2 N

sin x dx x

,G +P =%! G.P a > Q<_ < n< & g(x) = e−ax V+ i: = =%!

G d Tf (x) = sin x/x -" i: S T
e−ax

=%! G.P _+

-

-

a

∞

f (x)g(x) dx =

a

x

∞

e

−ax sin x

x

dx &d

,G d + ( P " " %! < G.P < M7; G.P TG.P " + )

) ) )

a

f (t, x) dt

,G T+

i: &

L S + i: PW cM C/I= " =%! [a; ∞) × S & 5& f (t, x) < R L +l C. %! ]KQG [a; ∞) & f (t, x) x ∈ S P ǱL& + i: B< [a; ∞) & M (t) 5& < =|f (t, x)| ≤ M (t) (t, x) ∈ [a; ∞) × S P ǱL& w[Q - ∞ =%! G.P M (t) dt w ∞

=

∞

∞

-

-

+x

-%! < G.P sin(x ) dx d √ %" Tx = x i: & L T7& =

=%! G.P

a

<

-$ =%+ M7; G.P T,G ^

M!

∈ [−; ]

−/

lim g(x) = lim

x→∞

V+hS 4

f (t, x)dt = F (x)

g (x) = −x( + x )

V+ %Q ^ ∞

sin x

- b ∀ ∃b ≥ a ∀b ≥ b ∀x ∈ S : F (x) − f (t, x)dt < a -

− sin x cos x dx

= sin x −

< +l S ⊆ R + i: " f (t, x) + i: =[a; ∞) × S = {(t, x) | a ≤ t, x ∈ S} x ∈ S P ǱL& < S [5 [a; ∞)×S & %! 5& . ="& G.P F (x) = a∞ f (t, x) dt ,< ! ,G

!0 F (x) ' D. . Q

x

)

T24 ^ 90

∞

-

∞

-

∞

-

x cos x dx, +x

)

cos x dx, + x

)

sin x dx, e x−

)

cos x dx,

)

∞

-

xp (ln x)q dx,

-

∞

-

∞

-

x sin x dx, +x

∞

cos x dx, −x

−∞

-

)

sin x dx, x(x − )

∞

-

xp sin x dx,

∞

xp (e−x − ) dx.

#*7; 4 *P4$ /#18 <@Q#:*8 2

/#18 "#:*8 2

!+H S & F 24 ^ =%! 88 G.P F (x) & =%! " " & -%! 5& f (t, x) + i: ∞ ǱL& f (t, x) dt ,G < " [5 [a; ∞) × [c; d] a i:
∞ ∂ f (t, x) dt ! ,G < 88 24 & [c; d] & ∂x a < %! ]H+ [c; d] & F 24 ^ =%! G.P . ∂ f (t, x) dt QG & & V8 =F (x) = a∞ ∂x =%! /& < !

(Q x < < r ǱL& + i: F (x) =

∞

2

t→

−rt e sin(xt) t −rt sin xt |x|e xt

f (t, x) = =

ke−rt = M (t) - ∞ G.P k/r & M (x) dt

&

[c; d]

&

∞

e−rt sin xt t dt

L < %! !+H

-

&

S = [c; d]

= =

∞

e−rt cos(xt) dt

r r +x

2

T24 ^

"

= tx− e−t ≤ td− e−t = M (t) - ∞ = td− e−t dt

& f ,G V+S + Tw-$x %! G.P =%! 88 G.P

,G G d T < G.P [a; ∞) & k

a ∈

S +P- ∞ N P ǱL& (x

2 N

+ t )−k dt

=%! 88 < M (t) = (a + t )−k V+ i: >=9=I = " + < a ≤ x L $ T24 ^ =S = [a; ∞) V S & mH Ta + t ≤ x + t

-

T
= (x + t )−k

≤ (a + t )−k = M (t) - ∞ - ∞ T%! G.P M (t)dt = (a + t )−k dt

L

& e−rt cos(xt) dt < %! e−rt cos xt && x & V R & mH Tw-$x %! 88 G.P r/(r + x ) F (x)

M (t)dt

∞

V+S + c "< =%! G.P F (x) & 88 24 & F V+S + F=9=I L

R

∞

f (t, x)

[c; d] %D e−rt sin(xt) h_B M $ V+S + I=9=I - ∞t

f (t, x) dt a

=%! 88 G.P [c; d] ⊆ (; ∞) @ & @L& P & i: & G d T ≤ t < < c ≤ x ≤ d S = 8 & B & < >=9=I M (t) = td− e−t

≤

-

a

b

tx− e−t dt ,G +P

lim

,G ' F (x) mH Tf (t, x) = G d Tx = S _+ < i: &
`M Ld L T%!

∞

f (t, x)

e−rt sin(xt) t t→ sin(xt) = e x lim =x xt t→

=

-

e−rt sin(xt) dt t

-

=%! 88 G.P S &

"

=+&+& F (x) @;& I=9=I L ! & x = P ǱL& V+ B p ^ & = lim f (t, x)

L S + i: PW cM C/I= " (a; b] × S & 5& f (t, x) < %! R L +l C. ]KQG (a; b] & f (t, x) x ∈ S P ǱL& + i: =%! B< (a; b] & M (t) U& < %! =|f (t, x)| ≤ M (t) (t, x) ∈ (a; b] ×-S P ǱL& w[Q b =%! G.P M (t)dt w

−k

(a + t )

dt

≤ =

∞

(a + t )− dt

lim

b→∞

a

arctan

b t π = a a

= P /& >=9=I s" ]Q L +l C. L S + i: ! " < %! [a; ∞) × S @C. & !+H 5& f (t,- x) < %! R ∞ ^ =%! 88 G.P F (x) & S & f (t, x) dt a =%! !+H S & F 24 f (t, x) < %! R L +l C. L S + i: S

9>

&

-

b

f (t, x) dt a

< %! [a; b] × S @C. & !+H 5&

/#18 "#:*8 2

#*7; 4 *P4$ /#18 <@Q#:*8 2

a, b ∈ [; ∞) P ǱL& + %&E -

∞

e−bx − e−ax cos x dx = x

+a +b

ln

2 N

V TF () = $ T+

= f (x) − F () - x = F (u) du - x r du = r + x

F (x)

V+ i: p ^ & = F (x) =

∞

− e−xt cos t dt t

r > P L & " %&E T^&& < -

i: & L T%! G.P %! %.! ,G f (t) =

t

−e

−xt

∞

x e−rt sin(xt) dt = arctan . t r

24 &

cos t , g(t) = e−xt cos t -

8 & B & < %D Ld f (x) t→∞ g(x)

=

lim

t→∞

P = + d G.P T

-

e

lim

xt

− t

xext

lim

t→∞

=∞ -

∞

∞

< =S h_B M 8 < I=9=I @+A/ '. & ^&& T"& e−xt cos t & && x & %D − e−xt cos t t V g(t)dt

-

F (x)

=

=

∞

e

−xt

e−xt cos t dt

Γ(x) :=

Γ(x + ) =

= = =

∞

cos tdt

= =

F (b) − F (a) b + ln a +

,G N Ta > + i: -

Ia,b :=

∞

e−ax

+ mK! < D!3 b < -

∞

a

b

2 N

tx e−t dt

#

4

lim

b→∞

-

5 x −t b

−t e

+x

$

b

t

x− −t

e

=

- b bx + x lim tx− e−t dt b→∞ eb b→∞ - ∞ +x tx− e−t dt

=

x Γ(x)

dt

lim

.∞ ) = e−t dt =

V+S + TΓ(

F (x) − F () - x x dx x + ln x +

e−bx − e−ax cos x dx x

b→∞

=

^&& < -

lim

=

+ F (x)

tx− e−t dt

@C & ^.x " %&E =x > TS [5 G.P [c; d] ⊆ (; ∞) @ & @L& P & Γ(x) w \< L ! & < (Q x > ǱL& T
x x +

=

∞

; 2 N

Γ( )

=

,

Γ()

=

× Γ( ) = ,

Γ()

=

× Γ() = ,

& B & mH

···

=Γ(n + ) = n! V n ∈ N ǱL& T. < w @C & ^. C &x " & Γ(x) M [c; d] ⊆ (; ∞) @- & @L& P &

∞ & 24 & =%! 88 G.P tx− e−t ln t dt Γ(k) (x) =

sin(bx) dx x

-

∞

√

G d Tu = t

[7 ( N L & b P L & +G&

Γ

π sin(bx) dx = sgn(b) x

k

S " < 7B -

= =

99

tx− e−t (ln t) dt

∞

-

t−/ e−t dt ∞

e−u du =

√ π

#*7; 4 *P4$ /#18 <@Q#:*8 2

/#18 "#:*8 2

& T+ < De < P a − b < a + b G d Ta > b S V D/ ,e V8 & -

Ia,b

=

=

∞

sin(a + b)x dx x - ∞ sin(a − b)x + dx x π π π + =

sin(bx)

& %D Ia,b T%! !+H b & %D e−ax $ = x + < %! ]KN -

dIa,b dt

=

Ia,a

=

π

=

∞

=

()

=

+ Ta − b < < a + b > G d Ta < b S O& < Ia,b

=

=

Ia,b

=

⎪ π ⎪ ⎪ sgn(a) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

= sgn(a)

π

=

=%! " ! Ǳ_B & Ǳ_B \< L w0x < w x

T+

S 2DE T. mH

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ π ⎪ ⎪ ⎪ ⎨ sgn(a)

=

∞

sin(a + b)x dx x - ∞ sin(a − b)x + dx x π π − =

a dIa,b = dt a +b

^&& <

< |a| < |b| S Ia,b = arctan

< |b| < |a| S < |a| = |b| S a = S + sgn(|a| − |b|)

"

TIa,

-

R

-

&

+ %&E T

sin(bx) dx = x

∞

cos(xt) dt +t

=

L & Ia,b

+ %&E w> =%!

-

Ia,b =

∞

e−x sin xt dx = t/(t + ) e−x ( − cos xy) dx = ln +y x

:=

∞

I,b = lim Ia,b a→+ b π lim arctan = a a→+

2+ N

sin(ax) cos(bx) dx N x =+ D!3 b < a [7 ( N

" B =

8 & B & < 1=9=I n< \< & w9 ∞

∞

88 G.P

-

=

∞

=F (x) = t− ( − e−xt ) dt + i: ≤ x ǱL& w0 =+&+& F (x) @;& T .C 1=9=I ,e &

b +C a

V b = L & $ =%! %&E C C

=Ia,b = arctan(b/a) + < C = ^&& ^&& < sgn(b) = - %! & G dT b = S ∞ π sin(bx) dx = sgn(b) < $ ^ + =V+G& a = + d %!&

,G G d T < a < ≤ k S - +P w ∞ tk e−xt dt =%! 88 G.P [a; ∞) & -

e−ax cos(bx) dx

∞ b ∞ −ax −ax sin(bx) e + e sin(bx) dx b a

∞ cos(bx) + −e−ax b b ∞ −ax − e cos(bx) dx a - ∞ a a − e−ax cos(bx) dx b b a a dIa,b − b b dt

=

sin(a)x dx x π

cos(bx) dx x

()

G d Ta = b S

xe−ax

-∞

=

∞

Ia,b

96

∞

sin(a + b)x sin(a − b)x + x x

dx

$ =a < b < a = b Ta > b V+S p %Q ! =a > V+ i: k / T%! : a & %D

/#18 "#:*8 2

#*7; 4 *P4$ /#18 <@Q#:*8 2

) B(p, q) = Γ(p)Γ(q) ,

-

Γ(p + q)

-

)

π/

sinp x dx =

-

) )

B

cosp x dx =

sinp x cosq x dx =

) B(p, q) = B(p +

p+

,

B

p

-

,

p+

,

q+

L {f (t)} :=

,

) L {

B(p, q) =

∞

xp− dx ( + x)p+q

+y

+a ++a & w>)

=

m > P L & +P − cos(mx) −x e dx = ln( + m ) x

∞

-

- n = x − ln x ∞

dx = (x + a)n+

-

= -

-

∞

xn e−ax dx =

∞

cos(mx) dx = +x

dx = ln(n + ) + %&E

n! an+

) L

) L

-

∞

-

x ( + x ) dx ( + x)

w>>

2

) L

(s > )

,

(s > )

k! sk+

ect =

,

(k ∈ N, s > )

,

s−c

,

(s > c)

tect =

(s > c)

(s > |a|) (s > |a|)

e−ct f (t) = F (s + c),

+ %&E w>6

) L {f (t − c)} = e−csF (s),

+ %&E w>F

) L {f (ct)} =

c

F

s c

,

) L {f (t)} = sF (s) − f (). t > - Γ(t) = ln dt t Γ

# "

(k)

(x) =

t

x− −t

e

k

(ln t) dt

p & T < q < < p L & + i: -

B(p, q) :=

x ( + x ) dx = ( + x)

P L & +P w0)

k ∈ N P L & +P w0

∞

8" & q

=+ D!3 V J=9=I L >) ^. '. & =

∞

e−st f (t)dt

G d TF (s) = L {f (t)} S + %&E

π x sin(mx) dx = e−m +x

N

, (s − c) )L √ =Γ , t ) L {sin(at)} = a , s +a ) L {cos(at)} = s , s +a ) L {sinh(at)} = a , s −a ) L {cosh(at)} = s . s −a

w>0

& _+ < S [5 1=9=I S U& '. & <& +& @+t+H QG T" [5 J=9=I ^. B L & =. D!3 TL $ @D!3 +

s

k t =

+ %&E w>9 π × × · · · × (n − ) n n!an+/ ∞

s

) L

∞

-

}=

) L {t} =

G d T"& %De C b < a S + %&Ew> (a + b)a+b arctan(ax) − arctan(bx) π dx = ln ab b a x

-

∞

S + %&E 24 ^ =wf (t) ^+!fHO + (&x

G d Ts >

),

+P x =

- y √ sin(xy) dx = π e−t dt x

+ i:

) B(p, q)B(p + q, r) = B(q, r)B(p, q + r). -

−x

B(p + , q − ),

p

e

,

, q) + B(p, q + ),

) B(p, q) = p + q B(p, q + ) B(p, q) = q −

,

π/

p+

π/

-

B

+ %&E 1=9=I n< \< & w6

∞

xp− ( − x)q− dx

+P 24 ^ =V+ [5 9F

- . / 0*1 '2

/#18 "#:*8 2

(b − a)p+q+

= ()

-

- ∞ x− x− = dx + dx ( + x)+ ( + x)+ = B( , ) + B(, )

θ (b − a) sin θ cos θ dθ

q

cos -

π/

p+

sin

θ cos

q+

θ dθ

(b − a)p+q+ Γ(p + )Γ(q + ) Γ(p + q + )

=

=

=%! " ! J=9=I L 06 ^. L w x

=

2+ N

=

< "& 4 [Q( C n + i: - Im,n,p :=

- m− x + xn− := dx m+n ( + x)

=

=

-

n -

=

y (m−n+)/n ( − y)p dy = (m+)/n−

(p+)−

( − y)

=

-

π/

-

∞

sin

m−

-

n−

θ cos

θ dθ

=

"

√

π

Γ

- In

w w0

n! m(m + a) · · · (m + an)

w>

=

=

∞

x

xm e−n

dx =

n

Γ m+

=" ! y = n =

-

∞

x

m+

+a ++a L

x ( + x ) dx = ( + x)

+, - ./(0

=

=

dx

-

√ π/

n - π/

dx − xn sin /n− θ cos θ dθ cos θ

sin /n− θ dθ n Γ n Γ n Γ + n

=%! " ! J=9=I L 06 ^. L w x

w9 w6

=

()

=" ! y = xa +a ++a L .P = -

2 N

√ =+ D!3 In := − xn n V x = sin θ +a ++a & =

Γ(n)Γ(m) m a bn Γ(n + m)

(a sin θ + b cos θ)m+n

.P =

y m− dy + y)m+n

,G N T"& %De C n 4

√

− x

xm− ( −xa )n dx = an

=

xn− dx + x)m+n

-

dx

dy

,p +

+P L < L ' P =

- xm− dx + m+n ( + x) ( - - ∞ m− x dx + ( + x)m+n ( - ∞ m− x dx ( + x)m+n B(m, n)

= =

y n m+ B n n

=

- Im,n

xm−n+ ( − xn )p (xn− dx)

2 N

,G N 24 ^ =+ D!3 V y = /x i: & =

Im,n

=+&+& Im,n,p N 24 ^ Tx = y V+ i: = Im,n,p

n

-

×

T"& %De N+N C n < m + i:

xm ( − xn )p dx

B(, ) Γ() Γ( ) Γ( ) ××× × × ×

=

(b − a)p+q+ B(p + , q + )

=

∞

C q < p < (Q C b < a 4 D!3

-

b

24 ^ Tx = a cos

%1

-

,G N T"& %De =+

θ + b sin θ V+ i: =

(x − a)p (b − x)q dx a

b

(x − a)p (b − x)q dx =

%3 W(& & TK+ _: L ! 2N @P & =" 5B ' j: L ^+.P

a

=

9I

-

π/

(b − a)p sin

p

2 N

θ (b − a)q

/#18 "#:*8 2

- . / 0*1 '2

< limit 2 ! '. & G ; A " QG < . ! ! ,G [5 L int . +& K+

@ K+ )4 G ; A " ^++5 ! QG %! 2 ! L 5+!< QG & int ! & ^&& N T QG K+ t $ =. D!3 !

! evalf ! L ^+$ = S & k+C d . & = D!3 d DN N +

K+ 1 √ √ int(cos(x ∧ 2), x = 0..inﬁnity) −−−−→ int(x ∧ 2sin(x ∧ 3), x

K+

=

4

F := b− > int(f(x), x = a..b) −−−−→

f (x)dx a

K+

limit(F(x), x = inﬁnity) −−−−→ lim F(b) b→∞

K+

− 1..inﬁnity) −−−→ - +∞ 1 2 x sin dx x4 1

. & K+

F := b− > int(1/(x ∧ 3 + 1)(x), x = 0..b) −−−−→ - b 1 F(b) = dx 1 + x3 0

evalf(%) −−−−→ 0.9819638240

K+

yD/ s @+ z 5 & % TK+ s+3 8 Y+ =%!

F(b) −−−−→ b −b+ ln b + b +

+

√

arctan

√

"

http://webpages.iust.ac.ir/m_nadjafikhah/r1.html

=%! "
b

F (b) =

2 π

cd

-

K+

limit(F(x), x = inﬁnity) −−−−→

91

2 √ π 3 9

(b −

) + π √

& & '()" n ≥ P ǱL & < x = x =

2+ N

Tx

2# N

=

Tx

$ D+: @QD =xn = xn− + xn− = π C / @QD = = = < x = Tx = Tx =

Tx

=

=%! QD {xn }∞ n=a + i: < %! C && < B QD

M! !0 C & {x }

@QD 4 TG +& & xn @74: %D/C ε > P ǱL & w Tx QD =w|xn − | < ε 5& & n L T5x " ε L .

=V+ T"D G.P ∞ n n=a

C D

2

!;!

@QD +P n =n→∞ lim / = %! G.P 4 & − log ε < n /ε < n 5 & | /n −| < ε " = ^ ="& − log ε && N V+ i: & mH =%! 24 ∞

∀ε ∃N ∀n

n > N ⇒ n − < ε

n + & n +

∞

2 N

@QD +P =%! G.P n + < ε 5 & − < ε " = (n + ) n + " i: %! : mH =%! − < n ε =

n=−

N=

V+hS 4 n→∞ lim xn = V+

∀ε ∃N ∀n (n > N ⇒ |xn − | < ε)

{ /n }n=

%N+N =%! ! < QD & "d j: ^ L vP V+P ^ nX & ,G < + Ok4 % d . 74 @ +L

,e C & = +H 5 V+.5 & ^+t.P =%! +& &/ PQD s! T"& e3& [5 &/ V4 U& T5& PQD & QD ="&

− ε

O7 Z

2& d '8 ( ) "

H

5& @C. T x Z ^+hH L @C. L ' L %! =R & w{a, a + , · · · , n, n + , · · ·}

@: . & R & {a, a + , · · ·} L x @QD {7l xa , xa+ , · · · , xn , · · · @ S . & < {xn }∞ n=a S n d & < xn = x(n) TP =" ∞ @C. L %! 2DC {xn }n=a

'()" F! '8P0U

'8()"

{xa , xa+ , · · · , xn , · · ·}

! ! P M7Z M): {x }

@QD V+hS 4 P ǱL & " %: N

="& P

∞ n n=a

%+4 xn n ≥ N

xn

= n

5+DX C @QD

2

={n}∞ n= q ≤ n q ≤ n xn = n + : C @QD ={n + }∞ n= n ∞ = ( /) n=− q− ≤ n xn = /n @QD Tx = Tx = yxn = ,< C ^+ nz @QD = = = < x = Tx =

2 N 2 N 2 N

9J

< <#1 $ Q 8 I

Txn =

√ n n−

n

Q 8 [ I

2# N

√ n

V+ i: S L Tn→∞ lim n= V T ≤ xn $ G d

= (xn + )n =

+ nxn +

>

+

n(n − )

n(n − )

xn + · · · + xnn

x

n √ TN = /ε < n S mH =| n − | = xn < /n ^&& √ =" %&E V8 < n − < ε G d n

2 N

=n→∞ lim an = G d T|a| < S + %&E T24 ^ +l =%! & V8 G d Ta = S = i: %! : ^&& =%! |a|n < ε 5 & |an − | < ε =N = − log|a| ε " &&

n

Tn→∞ lim yn = m < lim xn = + i: n→∞ )

)

lim axn = a

n→∞

S L =lim a a ∈ R P ǱL & n! n > m P ǱL & G d T"& [|a|] +

n a − n!

24 ^

lim xn yn = m

=

P L & S w6 xn lim = =n→∞

_+ < = = / V+ i: w0x 2DE & n ≥ N P ǱL & 24 ^ =N = max N , N V

=

N=

< <

√ n

lim =n→∞ xn =

|xn − | + |yn − m| + =

n a < − < ε G d n!

2+ N

< a P ǱL & +P V+S p %Q ! = √ + < < a G d T < a S w[Q ^&& =%! %De

a=

√ n a−

a

n

= (xn + )n =

+ nxn +

>

+ nxn

n(n − )

+ xn xn + · · · + nxn− n n

a− a− < n S mH =xn < ε n √ n a − = |xn − | < ε

G d TN =

√ n

=n→∞ lim

|xn yn − yn + yn − m|

|a| n

−(m)! log ε < n |a| m

=" %&E V8

|xn − + yn − m|

|yn ||xn − | + |yn − m| + M M [|| + ] + =

···

" i: S

=%! w0x +D" w>x 2DE ǱL & %P N < M V+ i: w9x 2DE & V+ i: w^.x |yn | < M n ≥ N P N = max N , N , N S = = < = (|| + ) M

G d n ≥ N < ≤

|a|

m +

m m m |a| m ··· (m)! m + m + n n− m m |a| (m)! n |a| m (m)!

m

∀ ∃ N ∀n n ≥ N ⇒ |axn − a| <

|xn yn − m| =

|a| m |a| (m)! m +

≤

& %P N T (Q > ǱL & i: & & A7S" > P ǱL & _+ < |xn − | < n ≥ N P ǱL =|yn − | < n ≥ N P ǱL & %P N T (Q a = %Q L Ta = V+ i: w x 2DE & N = N < = /a i: & 24 ^ =%! & V+ p f

<

= ≤

= ≤ m G d Txn ≤ yn %D/C S wF

≤

|a|n n!

n→∞

yn

|xn + yn − ( + m)| =

=

lim xn + yn = + m

n→∞

) n→∞ lim xn − yn = − m ) G d Tm = < yn = yn

2 N

n

m

√ n a=

+ <

%./ M&; + <

n − < ε T a

=n→∞ lim

6)

√ n a= <

a

=n→∞ lim a = ^&& <

G d Ta = S w

G d T < a < S w|

+ =n→∞ lim

√ n a=

^&&

= V w[Qx a √ √ = − n a < ε n a < ε n

Q 8 [ I

< <#1 $ Q 8 I

24 |_ |( < 24 & lim

n→∞

n +n+

−

lim n +n+

=

n→∞

lim

=

n→∞

√

=

√

+

n −n+

2+ N

V

P ǱL & N < M > V+ i: w6x 2DE & V+ i: =w^.x M < |yn | n ≥ N

=

" i: S , = = |m|M < ||

G d n ≥ N < N = max N −

n

n −n+

+

+ n + n +

− n + n

"

n

−n+ ) lim = n→∞ n + n +

)

n→∞

)

n→∞

+ + ···+ n = n nπ = cos

lim lim

)

lim

n + − n −

n→∞

=

+ D!3 L PQD L ' P ) n − n + ) n + n − n + n + n − n − n + n + n + n + ) + + · · · + n ) n + n + n + n + ) n +

) n n

)

√ n+

√ + n

)

)

) ) )

n

n

loga n ×

n

+ ··· +

⇒ yn < + =

+ n + n +

+ +· · ·+n =

V T lim

n→∞

=

2

lim

n(n + )

+ + ···+ n n

!

+ /n + /n n→∞ + /n ++ = +

= =

2 N

8 & B &

n(n + )/ n lim + = n→∞ n lim

n→∞

V 9= =1 ,e L w6x %./ & B &

√ n n!

lim

n − × ···× n n−

+

|yn − | <

+m

=

)

+m

n→∞

) n sin(n!) n+ n ) n

(a > )

|xn − | < ⇒ xn > − =

n

lim

√

) cos(n ) − n n n + √ √ √ √ n ) · · ·

V n & V+N &

√ n + + n ) √ n + n − n ) − n + n

√ n

, N , N

=%! A i: & yn < +m < xn +

2

n

) n − n + n

& mH T"D ≤ m S V+ i: wFx 2DE & −m N < N C T = = i: & , =m <

G d n ≤ N = max{N, N } S %P

n

M

xn mxn − yn yn − m = myn |mxn + m − m − yn | = |m||yn | |m||xn − | + |||yn − m| < |m|M || |xn − | + |yn − m| = M |m|M < + =

=

+P TQD [5 L ! &

=

√ n

n

n→∞

= =

lim

√ n

n→∞

×

lim

n→∞

√ n n

− + − · · · − n n + + n −

lim

n

=

lim

n→∞

+ + + · · · + n + + + · · · + n

6

n

n→∞

+

2 N

=

V T^ /S } |( &

n

− n + n +

2 N

=

n − n + n + n + n +

=

< <#1 $ Q 8 I

LQ 8 #:L <@8.a I

mH = − < xn < + − < xn − < + =O& L VP < %! ^+H L VP xn n ≥ N P ǱL & " i: t $

)

(n + ) − (n − )

)

×

+

×

+ ···+

n(n + )

=%! 8 B< 24 QD + %&E

A = max{x , · · · , xN } , B = min{x , · · · , x)N }

2%

n P ǱL & G d

M O7 !DM 9=7-`

+ A − < xn < + B +

=%! w>x V8 DN m8C w9x V8

2

Tx

=

√ √ a + a Tx = a

2

@QD

√

i< C < a xn = a + a + · · · + a < === %! G.P QD ^ + %&E =+G& p %! %&E < =+&+& d mK! < √ =xn+ = a + xn U/< d & QD [5 L = QD ^ V+P TQD ^ G.P 2DE & =%! O& L < 54

G d Txn+ < xn V+ i: S L T%! 54 QD a+

<

^&& =%! a + xn

< xn

√ a + xn < xn T xn −

+

a+

< xn <

−

a+

$ =%! 8"d / ' ^ T =

+

<

+

a+

<

−

a+

<

−

&7

4 {xn }∞ @ QD n=a 4 QD =xn+ ≥ xn n ≥ a P ǱL & V+hS & =xn+ > xn n ≥ a P ǱL & V+hS =%! [5 &/ Q<_ < @QD & 24

Y! ! ( S"

H >!

&7 Y! !

4 {xn }∞ @ QD n=a n ≥ P ǱL & S %: $ M C V+hS V+hS 4 QD =xn ≤ M a =M ≤ xn n ≥ a P ǱL & S %: $ M C = @QD T"& ^+hH < O& L QD

!"! J >!

T O& L < 54 @QD P w

!"!

=%! G.P =%! G.P T ^+hH L < Q<_ @QD P w0 =%! TG.P @QD P w> =%! S< T & @QD P w9

=

V+ i: w x 2DE & A7S" M&; C. ^ < "& P xn @.P O& ^ 8$

=%! H! ]Q < %! +l < O& L i: lim xn = a V+P mH ="D ^+$ V+ i: n→α n ∈ N ' N > P ǱL & %P < xn − a > 24 ^ =|xn − a| > B< $ P xn O& ^ 8$ a . mH =xn > a + =%! / "& L < Q<_ w0x %Q G d Tyn = −xn " i: S =%! yn O& L < 54 5 & xn ^+H =%! . P& w x %./ & & < i: & = = n→α lim xn V+ i: w>x 2DE =|xn − | < n ≥ N P ǱL & %P N ' T = a = sup{xn |n ∈ N}

< xn

i: mH

=xn+ ≥ xn G d Tb = max {, a} S L T%! O& L QD < xn > a 24 ^ +l < Txn ≤ b + < xn ≤ a + xn

<N QD ' M+/ N ^ : TU/ L +& L +.X L mH =. ^+.A d B< T%+ &x DN "< < 7+73 P\< & T B< =. 2D d N ^ : & w+K '.

n P L& ^&& < T%! s7l xn+

H

=

a + xn−

≤

a + xn ≤ xn + xn = xn

=xn ≤ b VP L& ]Q < xn < ^&& Txn > n< & $ < =xn ≤ b n P L& 5

60

LQ 8 #:L <@8.a I

+

< <#1 $ Q 8 I

]" QD ^

=%! O& L < 54 =V+P e . & < T^+ 7.B< ,: L ! & p ^ & = V+ xn

=

=

≤

=

+

n

n(n − )

+ ··· n n(n − ) · · · (n(n − )) ···+ n! n + − ! n − − + + ··· ! n n n− − − − ··· ···+ n! n n n + − ! n+ + − − + ··· ! n+ n+ ···+ − − ··· n! n+ n+ n− ··· − n+ + − − ··· (n + )! n+ n+ n − ··· n+ xn+ +n

i: TV {xn }∞ @ QD G.P L , n= + xn = a + xn ^ & B & < n→∞ lim xn = V+ V+S =

lim xn

n→∞

=

n

lim (a + xn− ) = a +

n→∞

= = ± + a − − a = ^&& T"& d . mH T De P xn @.P $ lim xn = 5 =%! < − + a + n→∞ = + + a

2 N

< x = b TP(Q %De C < b < a +

i: a x V+P =xn+ = √n + xn ≤ n P ǱL & p ^ & =%! G.P a C & QD ^ V+P =V+ %&E xn ^+hH L < & Q<_ ǱL & V+ U/< T%! ^+hH L %D/C xn + T%! /& h + xn+

= = ≥

xn

=

+ +

−

···+ ≤

+

≤

+

n!

n

− n −

+ ···+

!

+

!

+

×

=

n!

+ ··· +

+

√ a

+

+

=

− ( /) − / n− − ≤

n−

+

mH T De P

G.P xn @QD ^&& =%! ! C & O& L xn mH V+ = e C QD ^ N =%!

a xn a + √ = ( a) +

lim xn = V+ i: S =%! G.P QD T^&&

G d Tn→∞ V xn [5 L

× ×!· · · × " C n−

+ ··· +

n ≥ P ǱL & T5 L T%! Q<_ xn

= ≤

n

=

xn+ xn

− + ··· n n− − − ··· n n n

=

a xn + xn √ √ a xn a √ + a xn √ √ a ×= a

!

!

≥ < T h %De C P

=xn ≥

h

lim xn+ a = lim xn + n→∞ xn a + = l √ xn @.P $ = = ± a = + a √ √ = lim xn = a ^&& < %! < − a n→∞

n→∞

Te =%%*. . . ]" 2 N

xn

6>

+ %&E =+G& p

n

xn = ( + /n)

QD

< <#1 $ Q 8 I

LQ 8 #:L <@8.a I

. mH Te ≈0I I1

' P + %&E O& \< & d 8 24 = P G.P L PQD L + D!3 xn− ) xn = , a + xn−

)

xn =

n n −

)

xn =

n (n + )!

)

xn = +

)

xn =

)

xn =

)

xn =

)

xn =

,

!

×

!

× ···×

+

+

+

+P Tex

+

−

xn =

+

−

+

lim

n→∞

xn

−

+

···

n+

+ ···+

2 N

n+n

=

n +

=

(n + )(n + )

x n

n

xn :=

xn

=

≥

>

=

n

>

n+n

+ ··· + = n ! n"

+

m+%%+ ( ! 2+ N

γ= +

+ ··· +

+

− ln n

+ ···+ + − ln n n− n - dx dx + + ··· - n - n dx dx ···+ + − n − n n− x - - n dx dx dx + + ···+ + − ln n x x n n− x +

n

≥

L T%! Q<_ QD ^
k

2

=

-

k ∈ N P ǱL & + i: ! @QD L ' {xn }αk= @QD 24 ^ =nk < nk+ =V+ {xn } @QD L @QD L QD L ' T,< C @QD

+ ···+

−

( !

L & %! $ {xn }∞ @ QD + i: w 0 n= +P 24 ^ =n→∞ lim nxn = d n lim ( + xn ) = e

! ^. L .P =n→∞ =+

n

n+

n +

n =%! ^&& < T%! ^+hH L < Q<_ QD ^ =P γ . & < + L T%! ^+hH L xn @QD =

4 w

< a + i: = ln a + ln b

n

@QD +P

n x −n =n→∞ lim − = ex n

+P 24 ^ T < b < √ =n→∞ lim n ab −

+

n+

+

C n V >=>=F ^. L w6x %./ T

n

+

= <

n + n

···

:=

+

n+

xn+ − xn

n +

+ ···+

+

xn =

n+

n −

n

V+ i:

n!

+ ···+

n→∞

n! nn

xn =

+ ···+

+

)

)

+

=%! G.P {xn }∞ @ QD + %&E n= L < 54 QD ^ V+P p ^ & = %! O&

n>

+

x = a >

n

e := lim

=

n+ - n+

= ≤

"

n - n+ n

=%! 5+DX C

=

69

− ln(n + ) + ln n dx n+

− ln(n + ) + ln n

dx − ln(n + ) + ln n x n+

[ln |x|]n

− ln(n + ) + ln n =

LQ 8 #:L <@8.a I

< <#1 $ Q 8 I

=|ym −a| < /m m P ǱL & TǱN ! & & mH ="& xn ∞ 2 =%! G.P a & {xn }∞ @ QD L T^&& n= L {ym }n=

{x } @QD V+hS 4 T%! @ '8()" ' j & < C n7 @ i@ ∞ n n=a

n≥N

P ǱL & S %: N ' Tε > P ǱL &

=|xn − xm | < ε m ≥ N P <

=%! " TG.P @QD P w =%! G.P T" @QD P w0

& ^&& =n→α lim xn = V+ i: w x 2DE & A7S" n > N P ǱL & %P N ' > P ǱL ^ =m ≥ N < n ≥ N V+ i: , =|xn − | < / 24 |xn − xn | = < <

|xn − + − xm | |xn − | + |xm − | + =

=%! " {xn } @QD mH =%! " @QD ' {xn } V+ i: w0x 2DE & n ≥ N P ǱL& %P N ' T = ǱL & @QD 5 = +xN < xn < +xN 5 =|xN −xn | < = L ' T @QD P +A/ & & =%! {xn } @QD V+ %&E %! : = {xn } G.P @QD =%! G.P {xn } %! G.P .P & {xn } %P N ' > ǱL & {xn } & " i: & & ' T^+t.P =|xn − xm | < / G d n, m ≥ N S

|xn − | < / k ≥ N P ǱL & %P N k P ǱL & =%! {xn } @QD L V m ≥ N < N = max{N , N } i: & mH =nk ≥ k k

k

k

k

|xm − | = ≤ <

|xm − xnm + xnm − | |xm − xnM | + |xnm − | + =

2

' [5 && =C C x x α = ± x + +

2

=%! . P& <

8& %! DC α C + ··· +

xn

+ ···

n Y+34 C G.P = = = < xn < = = = Tx Tx d

P G.P = = = < xn < = = = Tx Tx < ≤ x T P

9:

L QD L '

{ /(n + )}n= ∞

2

2 N

@QD =%! {

∞

/n}n=

T"& G.P & {xn } @QD S =%! G.P & _+ {xn } L QD L P G d {xn } @QD G d T"& S< {xn } @QD L QD L S =%! S< _+

2

: mH =%! w x +N m8C w0x %! ^"< A7S" @QD L ' {xn }∞ k+ V+ i: =" 2DE w x %! ǱL & %P N ' (Q > P ǱL & ="& {xn } k P ǱL & Ti: M&; $ =|xn − | < n ≥ N P k ≤ nk k P ǱL & mH ≤ n n< & < nk < nk+ < nk ≥ N k ≤ N P ǱL & + wN ! &`^.x 2 =%! G.P & _+ {xn } @QD 5 =|xn − | < ]Q k

k

2T

k

L =%! S< {(− )n }∞ @ QD n= < %! G.P ' C & d L x n = (− ) n = @QD ' { C & d L x n+ = (− ) n+ = − @QD L ?+P & xn = (− )n mH = − Q =%! G.P =%+ G.P

2 N

L TL =%! G.P 4 & { /n!}∞ @ QD n= G.P 4 & %! { /n}∞ @ QD L QD n= G.P ="& QD L @QD P

#

=%! G.P

mH T"& P A S =A = {xn |n ∈ N} V+ i: A7S" xn = a V n ∈ N % & ǱL & %P a C P& < %! 2f.B ^ L 4 @QD {yn}∞ V+ i: n= A Ti: M&; =%! P A V+ i: mH =%! . ^ 8$ x ! H! mH =%! O& L < +l =a = sup A V+ i: =%! wO&

%P n T! H! [5 & & T = ǱL & T = / ǱL & ="& xn d y V+ i: =a − < xn < a a − / < xn < a %P n T! H! [5 & & V+ i: ǱN ! & ="& xn d y V+ i: =xn = y < ' T = /m ǱL & 24 ^ ="& " ( ym− =a − /m < xn < a n ≥ N P ǱL & %P N 8 /O P xn ^ ^+& L %! P P xn 5 $ d ym V+ i: = r: y , y , · · · , ym− & %P 66

< <#1 $ Q 8 I

LQ 8 #:L <@8.a I

@QD V+P xn =

+

+ ···+

2 N

n

=%+ " xn V+P p ^ & =%! S<

G d Tm = n S L |xm − xn | =

n+

>

m

+

/ ' ". 8$

n+

+ ··· +

m

)

xn = xn =

+ −

<x

+

!

+

!

|ym − yn |

− ···+

L ε " " 5 =%!

= <

= b Tx = a T < w <

+ ··· +

xm

m n+

+ ··· + m n+ m−n

− ( / ) − / n+

= n n+ − /

!

− ···+

2 N

(− )n n!

" ] @QD +P =+G& p T ≤ n

="& G.P + < T%! V+ B p ^ & = m n j ) (− )j ) (− ) − |xm − xn | = j! j= j! j= m ) (− )j = j!

T" {xn }∞ @ QD +P = ≤ n P ǱL & n= ="& G.P + < 5+DX C P L & C W. wF C W. & V8 = ! d TV8 ^ 24 +& ^. = B< k @H & =w+ 5B >=0=1 L w x %./ &x + %&E

j=n+

< k

≤ ≤

m ) j=n+ m ) j=n+

n ≥ P ǱL & < x = a > + i: wI L N ǱL & + M+N3 =xn+ = + /xn =%! G.P {xn }∞ @ QD a n= @QD " %: $ C C V+hS − xn+ | ≤ C|xn+ − xn | n ≥

n∈N

,

@QD

xn+ = wxn+ + ( − w)xn

= <

∞ n n=a

P ǱL & < < C < =|xn+

xn+

xn = − +

+ i: w6

n

+ T n < ε |xm − xn | < ε 4 TmH < %! " {yn }∞ @ QD ^&& =N = − log ε < n n=a = B< α C C 8 + ="& G.P +

n

(− )n (n)!

xn

x xm n+ = ± + · · · + m n+

≤

+ ···+

o)KC 4 {x }

≤

/

+ ···+

G.P {yn }∞ @ QD T5 B< α V+ %&E n= ∞ " {yn }n= QD V+ %&E Tp ^ & =%! %!

m−n=n

@QD +P P "& G.P + < %! " " sin sin sin n ) xn = + + ···+ n ) xn = cos ! + cos ! + · · · + cos n! × × n(n + ) )

x x yn = ± x + +

+ ··· + m ! m"

C

n = m

=

+

N+/ +& & =wN75 {, , , · · · , , } @C. & T5x d α = n→∞ lim yn

j!

j−

m−n −

n − / n−

n− > ( /)n− < ε |xm − xn | < ε 4 mH =N = log/ ε + " i: %! : ^&& ==log/ ε

6F

LQ 8 #:L <@8.a I

=

n+ n+ −

=

−

=

−

n+ − n+ −

n+

=

^&& < %! " TDN @QD P

n+

n+

≤

n+

n+

−

< <#1 $ Q 8 I

n+

−

+ ≤

=

n→∞

()

= =

lim

N →∞

C|xm− − xm− | + C|xm− − xm− |

≤

n+ −

lim

|xm − xm− | + |xm− − xm− | + · · · · · · + |xn+ − xn |

C

(xm − xm− ) + (xm− − xm− ) + · · · · · · + (xn+ − xn )

|xm − xn | =

n+

#

· · · + C|xn − xn− |

== =

n+ N $− + N

C m− |x − x | + C m− |x − x | + · · ·

≤

· · · + C n− |x − x | C n− − C m−n A< A C n− −C −C

e − < =

=N = −(n + ) %! " i: w x V F= =1 ,e L w0x %./ lim

n→∞

+ + ··· + n = n

2 N

S 5 T C

T5x + ' & d 2DC

n =%+ DN p @QD T^&& =n→∞ lim = n+ =

= |x − x | d

ε( − C) + < n − xn | < ε G d TlogC A =%! " QD ' {xn }∞ @ QD n=a

−C

A < ε

S mH =A

2

@QD "

n xn = + + + · · · + n! 24 ^ =+G& p (n + ) − xn+ | (n + )! =

|xn+ − xn+ | = xn+ − xn + · · · + (n + ) + · · · + (n + ) − (n + ) (n + ) = + · · · + (n + ) + · · · + n − (n + ) n (n + ) (n + ) (n + ) − (n + ) = (n + ) (n + ) − (n + ) n n n(n + ) = = (n + )(n + ) n+

n−

+ < |xm

+ T^&& =%+ DN ] @QD Q DN QD TG.P @QD P <_Qz V+S y="&

wC

="& G.P V+ i: A7S"

n ≥ a P ǱL & < < C < Tn < m S T24 ^ |xn+ − xn+ | < C|xn+ − xn |

G d

− n+

!

|xn+ |xn+ − xn |

= = ≤

(n + ) (n + )! n+ (n + ) n+ n(n + ) + n+ = n(n + ) + n

|xn+ − xn+ | < C|xn+ − xn | G d Tn ≥ S mH =%! G.P + < " TDN QD ^ 5 =C = =+G& p xn =

|xn+ − xn+ | = |xn+ − xn |

@QD +P P "& G.P ^&& < %! DN " 6I

2 N

+ · · · + n! QD @ nn 24 ^ (n + )! (n + )n+

+

(n + )! (n + )n+

< <#1 $ Q 8 I

LQ 8 #:L <@8.a I

' {yn }∞ / # n=a + i: ∞ $ {xn }n=a @QD < %! +∞ & S< < 54 @QD ∞ xn+ − xn %! ="& G.P α C & t $ =%! G.P α C P S< _+ xn /yn @QD yn

& <

/xn

2

Tn→∞ lim xn

yn+ − yn n=a & _+ xn /yn @QD 24 ^ ∞ x −x T"& S< n+ n yn+ − yn n=a

=&

L xn & TD/ @+A/ %! :

=" !

= a

V+ i:

2

A7S" /yn

zn+ − zn = lim x = a, yn+ − yn n→∞ n+ lim yn = lim n = +∞

x + x + · · · + xn zn = lim =a n→∞ yn n

G d Tyn = n! < xn = n i: & n = lim xn lim

lim

n→∞

=

lim

n!

+

+

(− )n

n

+ ···+

n +

xn − xn+ xn = lim n→α yn − yn− yn

=P $ < "& P $ n n+ =L < ∞ n→α lim = L V+ i: A7S" yn − yn− P ǱL & %P N ' > P ǱL & 24 ^ xn − xn+ X & − L < n ≥ N

x −x

xn − xx+
x

= = = Tn + Tn ǱL & w =1 x ,: =yn > yn+ L & TV+ U.B VP & d ^+:X < V+ n + p − < %" V+P T{+ ^

lim

n→∞

=

lim

& < V+P + % & & p < :S ^+:X L V
n

n × n!

n→∞

(L − )(yn − yn+p ) < xn − xn+p < (L + )(yn − yn+p )

(n + )! − n!

lim

n

n→∞

× n→∞ lim

n

n!

n

n

n!

n!

=

V 0)=0=1 yn = n < xn = ln n i: & = = = =

xn n→∞ yn x − xn lim n+ n→∞ yn+ − yn ln(n + ) − ln(n) lim n→∞ (n + ) − n n+ lim ln = n→∞ n

(L − )(y − ) < xn − < (L + )(yn − )

+

=

lim =n→∞

ln n n→∞ n

2 N

n+ − n

=

lim

+

− ··· +

yn − yn+

yn xn+ − xn lim n→∞ yn+ − yn

=

n→∞

xn =

+

(n!) (n)!

n→∞

=

n

)

lim

n→∞

n!

−

xn =

& G.P PQD {yn } < {xn } S 24 ^ T"& Q<_ QD {yn} < "& 4

V+S +

n→∞

xn =

lim

lim

)

n!

n→α

n→∞

n→∞

n

)

L

x + x + · · · + xn & =%! a && _+ n→∞ lim V+P n =xn = x + x + · · · + xn < yn = n V+ i: p ^ 8 & B & < 0)=0=1 && 24 ^

n→∞

) xn =

k_Q TmH

2 N

L − < xn /yn < L + V < yn $ T^&& < =%! L & G.P _+ {xn /yn} mH =|xn /yn − L| < @QD T5 =%! %+& L = ∞ V+ i: , ǱL& mH = + % & & {(xn − xn+ )/(yn − yn+ )} n ≥ N P ǱL & %P N ' M > P yn > yn+ $ T+ =(xn − xn+ )/(yn − yn+ ) > M xn − xn+ > M (yn − yn+ )

lim

n+p−

w0=1 x

< = = = Tn + Tn ǱL& w0=1 x ^+:X _+ & ^ ^&& =V+ U.B VP & < " xn − xn+p > M (yn − yn+p )

& xm > M yn " + p → ∞ & ^+:X L +S & 2 =%! % & & S< _+ {xn /yn } ^&& =xn /yn > M 61

LQ 8 #:L <@8.a I

p xn

< <#1 $ Q 8 I

2

= n /n

@QD L T%! G.P xn 24 ^ T+G& (n + )

lim

n→∞

xn+ xn

=

=

xn+ n→∞ xn lim

lim

n→∞

n+ n

<

2 N

@QD L %! S< QD T24 n

xn = n /n!

(n + )n+ (n + )! lim nn n→∞ n! n n+ lim n→∞ n n lim + n→∞ n e>

=

= = =

V 0>=0=1 @+A/ L ! %./ '. &

=

lim

n

n→∞

2 N

(n + )n+ (n + )! = lim nn n→∞ n! n n+ = lim =e n→∞ n

n→α

= =

xx+ xn

lim

n→α

−n−+(−)

−+ n→α lim

(−)n n

=

lim

n→∞

)

lim

p

p n→∞

+ p + · · · + np n − np p+

=

n+

−n−+(−) −n+(−) # "& : n S − "& |
=

+ p + · · · + (n − )p p = p+ p+ n

ǱL & mH || < < n→α lim n+ = $ w[Q A7S" xn ǱL & %P N Y+34 C || + < (Q x

>

xn +

^&&

− <

n ≥ N P xn x =|xx+ | < k|xn | n+ < k G d k = + || " xn

G d n ≥ N S

|xn | < k|xn− | < k |xn− | < · · · < k n−N |xN | =

2 N

/n

4 [Q( 2f.B & QD xn S x T24 ^ =n→∞ lim n+ = < "& xn < %! 4 & G.P xn G d T < S w[Q ="& %+& & S< xn G d T > S w x B n→∞ lim n+ < & %De 2f.B & QD xn S w| xn √ lim xn G d T"& =%! && 7D/ & < B< _+ n→∞ √ ^ =n→∞ lim xn = < "& %De QD xn S T24 < %! 4 & G.P xn G d T < S w ="& %+& & S< xn G d T > S wP

B 24

|xN | n .k kN

=n→α lim xn = ^&& < lim k n = mH k < $ n→α x > ǱL& =|| > < n→α lim n+ = $ w 2DE xn n ≥ N P ǱL & %P N ' || − >

a |x | mH || − < n+ < || + ^&& = n+ − < |xn | an P ǱL & < < k 24 ^ Tk = || − V+ i: S ^&& =|xx+ | > k|xn | n ≥ N |xn | > k|xn | > k |xn− | > · · · > k n−N |xN |

n+

=

)

+ p + · · · + np = p+ np+

i: S 24 ^ =

√ xx+ Q< "& B lim n xn %! ^8. n→α xn n→α −n+(−)n ^ =xn = V+ i: fke L ="& √ n xn

n→∞

lim

n

nn n!

lim

lim

p

)

n

=%! S< 0>=0=1 k+A/ && T+ <

n lim √ n→∞ n n!

@QD G d Ta > S +P w =%! +P L < L ' P

=%! G.P 0>=0=1 k+A/ && T+ < ^ =+G& p

n a

n

n

G.P 4 &

n+

lim

n→∞

=

|xN | =n→α lim |xn | = α ^&& < lim k n = α =|xn | > N k n n→α

n

k

wx 8 2DE =%! | ^ @74 L w|x 2DE 2 =" K! & < %! wx < w[Qx +D" wPx < 6J

Q 8 4 345 (47 I

< <#1 $ Q 8 I

/& & : < LO " ∞ G.P {xn }n=a @QD P ǱL & %! ^ x→x lim f (x) = ="& G.P & {f (xn )}∞ @ n=a QD Tx &

= B< n→α lim

# lim

lim n sin

n→∞

=

n

lim

=

√ n lim n cos − n→∞ n

=f (x) = cos x − x

< xn

= √ n

√ n lim n cos = − n→∞ n = =

& L =n→∞ lim lim

n→∞

n

lim

n→α

=

= =

n

+ αn =

+ αn

= =

n

)

)

=%! G.P _+ =%! G.P _+ =%! G.P _+

xn =

(n!) (n)!

)

/xn

+ 3& n

xn =

)

)

)

xn

G d T"& G.P

3

G d T"& G.P

3

|xn | G d

T"& G.P

3

xn /n G d

T"& G.P

3

n (n + )(n + ) · · · (n) = =n→∞ lim n e

2 N

lim ( + αn ) n→∞

un /n lim ( + un )/un n→∞

√ n

=%! S<

G d T < α < S V un = αn i:

TD

n

" n; C +Q u & TI 9 L ' P +P d & s7l T,e ' @h < 2DE

cos(xn ) − lim n→∞ (xn ) cos x − lim x→ x −

/n

L PQD L ' P G.P

V+ i: =>=1 ^&&

n $/

n+ n

n / n+ ··· = n n+ · · · n+ n+ lim n n→α · · · n+ n n+ n+ lim n→α n+ n+ lim + =e n→α n+

) xn =

2 N

···

N @D!3 &

V 0>=0=1 L w|x %./ & &

n

sin x = lim x→ x

=+ D!3

sin n

n→∞

n→α

2

^&&

2+ N

N

P ǱL & T^&& =x→x lim f (x) = V+ i: A7S" |x − x | < δ x P ǱL & %P δ > T > G.P QD {xn } V+ i: , =|f (x) − | < V P ǱL & %P N ' δ > ǱL & mH =%! x & n ≥ N P ǱL & ^D7& =|xn − x | < δ n ≥ N =%! G.P & {f (xn )} ]Q < |f (xn ) − | < Tx & {xn } G.P @QD P ǱL & V+ i: , ws7l i:x V+ i: ="& G.P & {f (xn )} @QD lim f (x) = P ǱL & %P > mH ="& s7l x→x =|f (x) − | < < |x − x | < δ %P x ' δ > |xn − x | < %P xn mH =δ = n < n ∈ N V+ i: %! G.P x & {xn } {+ ^ & =|f (x) − | < < /n {f (xn )} mH =". L . L f (xn ) @74: Q 2 =%! i: vf %+ G.P & N @D!3 & lim n sin ( /n) < xn = /n V+ i: =>=1 Tn→∞ ^&& =f (x) = sin x/x

xn+ xn

xn

S w9

xn

S w6

xn

S wF

xn

S wI

+P w1

O7 3 234 &36

H

QD _+ < N+N +a & +a ' U& ' 5Q; < d + @;& W(& ^ =V 5Q; =V+ F)

<#1 I

< <#1 $ Q 8 I

)

lim

n→∞

)

) )

n− n+

n − n +

n − −n lim n n→∞ + −n lim

lim tann

(− ) n

csc

lim

n→∞

√

+n

π

π

n→∞

) n

+

n→∞

)

n+ n −

n

n

lim

n

+

n→∞

αn

n→∞

lim sin ( /x) +P

=

x→

2+

lim ( + u)/u u→

=

^ =V+ D!3

lim m (n + a ) · · · (n + am ) − n

= B<

=>=1 M&; + Tn→∞ lim un =

+

exp

αn n→∞ n

lim αn /n n→∞

lim

αn n→∞ n lim

N %! : mH L T%! 4 &&

αn ≤ lim αn = n→∞ n

≤ n→∞ lim

="& e = && p T+

9!0

H

2 N

= B< x→∞ lim sin x V+P yn = nπ + π/ < xn = nπ PQD p ^ & = 24 ^ =V+S p

= P PQD L V +& < Z ! wP " x ! TV S PQD t d P T^&& ="& Y+34 _+ ! C L QD + i: i< @QD k h_B . T k ∞ {xn }n=a

[5 Sk x

( '8()" & ∞ )

<

xn

Q

P ǱL & =%!

≥ a k )

j=a ∞ {Sk }k=a

xj

24 &

. & < + wxn .C @7.B

= B< yn =

=V+P

xn

. & _+ d N T"& G.P ∞ )

xn = lim

n→∞

n=a

n )

lim tan ( /x)

< xn

nπ + π

xn

N+/ +& & =V+P

k=a

lim {xa + xa+ + · · · + xn }

L T%! G.P & ∞ ) n=

n

n=

=

n

!

lim

2

P N Tb > < a > 4 + D!3 L < L ' π n + n − ) lim ) lim n tan n→∞ n − n + n→∞ n

)

n )

n→∞

k=

k n+

− ( /) = lim n→∞ − ( /) n = − lim = n→∞

PQD p ^ & = 24 ^ =V+S p

nπ

="& B . p T =>=1 M&; + <

n→∞

∞ )

2+ N

+P

lim xn = lim yn = , n→∞ lim tan = , n→∞ xn lim tan = , n→∞ yn

xa + xa+ + · · · + xn + · · · = =

=

=>=1 && T+

n→∞

n=a

xk

= .

n→∞

x→

mH T%! QD ' ! P $ &&) ! S =%S ^(! d S< < G.P n=a

lim sin(yn ) = lim

= B< x→∞ lim sin x

xa + xa+ + · · · + xn + · · ·

= ,

n→∞

n→∞

@QD =V+

∞ )

lim sin(xn ) = lim

n→∞

n=a

∞ )

n→∞

= xa + xa+ + · · · + xk =

∞ {xn }n=a

lim xn = lim yn = ∞

n→∞

) )

F

lim n

n→∞

lim

n→∞

lim

n→∞

√ n

−

√ √ n n a+ nb

cosn √

n

) ,

lim

a−

n→∞

)

n→∞

)

lim n

lim

n→∞

n+ n+ √ n

n √ n b

+ a n

−

√

n+

< <#1 $ Q 8 I

<#1 I

+ + + + ··· + + · · · + ···+ n n n ! "

C

=

+

+

+

! C

∞ ) n=

n−

+ ···+

=

"

+

n(n + )

n

=

=

lim

=

∞ )

(− )n

n=

!

2+ N

. r4 " " d h_B

=

−

lim

n→∞

n )

Sn =

)

n=

n (n + )

∞ )

)

e−nα cos(nα),

−

− ··· +

+

n ) + + + · · · + n −n )

×

)

×

××

+

+ ···+

××

∞ ) n+− n+

) )

+

+ ···

lim

n→∞

(n − )(n + ) + ··· + +

+ ···

n(n + )(n + )

√ n

2 N

!

+ ···+

n!

> Sn

(n + )!

+

≤

+

+

+ ··· +

+

+ ··· +

+

− ( )n

n!

n−

<

−

+

+

!

+ ···+

= lim

n!

n→∞

∞ ) n=

n

n

+

'+ P !

=e

n

2 N

n

∞ ) (− )j j+

j=

n=

n!

h_B PC. @QD V >=0=1 ,e L w>x +Q =%! S< ^&& < . r4 " " d
n(n + )

) )

∞ ) n=

k

+ +

k

= =

∞ ) k k=

n

) k=

- P S< '+ -G.P L ! L '

)

=

%./ L =%! S<

q sin α + q sin(α) + · · · + q n sin(nα) + · · ·

∞ )

n=

w @C & ^. C &x

n=

)

∞ )

"& O& L <

=

+ ···

k+ =

n+

(α ∈ R) (− )n

54 QD G d

n=

)

k=

=+

k!

k=

Sn

n −

$

n )

S TU/< =%! O& L < 54 d h_B

Sn+ = Sn +

∞ )

−

k

k=

k+

PC. @QD L T%! G.P

n n+ ) ) k k (− ) − (− ) k= k= (− )n+ =

+ D!3 L ! .

2 N

−

k

k= # n )

n→∞

!

k(k + )

k=

n )

n→∞

= + %+& & d 2DC Tn → ∞ /<

|Sn+ − Sn | =

n→∞

lim

n(n + )

n=

n )

= lim

n

C. @QD L T%! S<

∞ )

L T%! G.P ' &

+

+

+

+

+

+

+ ···

··· + + n− + ···+ n− + + ≥ + + +

n

n!

F0

n

@ #1 #:L <@8.a 'I

{Sn }∞ n=a

%! d

∞ )

< <#1 $ Q 8 I

! G.P & : < LO

xn

n=a

@QD L =%! S< Sn

∞ )

="& O& L √

n=

2

!

n

+√

=

n√ = n

=

)

∞ ) (n!) (n)!

n=

+

G.P

n=

qn

D!3

2 N

+ q + · · · + qn − q n+ −q

=

∞ ) P (n) n a n!

n=

≤

/< < /<

xn

=!0 !DM 9=7-`

2/+ :K C/I=

=n→∞ lim xn = %! d

−q

!

# lim xn

=

n→α

=

m > n P < n ≥ N P

k=n

TO& 2< & " B %! : A7S" =%! O& L < 54 QD

T < a ≤

∞ ) n=

na

!

2

!

S TU/< = . r4 " " L T%! S<

G d Tm = n m ) = ka k=n

≥

n ) k=n

lim

n→α

=

k=n

n

V+ i:

L T%! S<

!

A7S"

$ xi

i=a

n→α

i=a α )

n− )

xi

i=a

xn

n=a

α−α=

∞ )

∞ )

(− )n

n=

qn

n=

!

2

!P ! G d T

≤ |q|

S

)n

2 N

=n→∞ lim q n = 24 ^ L

∞ )

" G d TSn

= B< fk4 n→∞ lim (−

xn

S

( )4-' C/I=

n=a

F>

n− )

xi − lim

xn −

xn

n=a

=%! . P& {+ ^ & <

& ! ' =

α )

xi −

i=a n )

n→α

2

k

n )

lim

n )

n=a

n ) k=n

≥

=

T%! S<

ka

xn

∞ )

n=a

n=a

2

α )

24 ^ =%! α & G.P

2W/' C/I=

m ) = xk < ε

%H

V+N +l "< L _+ ! %Q TPQD 3& .P G.P 5Q; ! dz W!H ^ & !H & = Ld "< ^ =" ! y-+ "& 5 T^&& q%! L +& Ld ^ 5 =V+ n; Ld ^ . L

ǱL & " %: N ' T < ε P ǱL & %! G.P

{Sn }

2% =+

="& O& L ∞ )

+ ···

! N T"& (Q N+N C

G.P & LO "

q n+ − −q −q

=

+ ··· +

+

√ n

@QD L T%! =

(n − )(n + )

n

!P ! G d T < q < S

Sn

n=

(n − ) n + + + ··· + + ··· n −

="& & O& L ∞ )

∞ )

)

' a < & k @B 7.B $ ' P (x) S P

+ ··· + √ n

C

)

√ + √ + ···+ √ n n n ! "

≥

)

= xa + xa+ + · · · xn

< "& 2f.B

< <#1 $ Q 8 I

<

∞ )

+ i:

xn

@ #1 #:L <@8.a 'I

L T%! G.P T

(43 $ C/I=

n=a

xn < B n→∞ lim < "& %De 2f.B & ! < yn

∞ )

xn yn

S =%! 8 ∞ )

S =& P G.P _+ _+

yn

∞ )

<

n=b

G d T"& G.P

xn

G d T"& G.P

yn

∞ )

∞ )

n n ( + /n)

n=

lim

n→∞

=

+

m ) ka

<

≤

xn yn

n→∞

=

n→∞

+

n→∞

n $−/

n

n + n + n − n= =

n

=

L w0x %./ &&x

! '

∞ )

xn

+ T <

n

lim

n→∞

= e−/ ∞ ) n n=

n

! <

∞ ) n=

<

lim

n→∞

+ n + n − n

∞ ) n=

n/

∞ )

n=a

xn

yn

<

∞ )

a

= .

+ i:

xn

$ C/I= "

n=a

n=a ∞ )

="& S< _+

yn

n=b ∞ )

G d T"& S<

n=a

L T%! G.P

=

! <

=w <

=%! S< F=6=1

2

n + ! n + n= <

n + n n

Lx %! G.P

! G.P & : < LO " T24 ^

∞ ) n=

="& 4 & G.P < Q<_ {|xn |}∞ @ QD %! n=a

n

n P ǱL & V+ i: A7S" V+P 24 ^ =xn > < %! 4 lim xn Txn ≥ xn+

=

n

∞ n ) n=

n −

!P ! <

F9

!

2 N

= √ < n n −

=%! S<

n→α

S w

H5;[ C/I=

L T%! S<

xn

n=b

∞ )

n + n +

= P < %De + 8 P xn 5 T%! ! d

lim

n→∞

$ <

<

V+"& " n ≥ N P ǱL & < "& %De 2f.B & !

G d T ≤ xn ≤ yn ∞ ∞ ) ) ="& G.P _+ xn G d T"& G.P yn S w[Q

n + n lim n→∞ n + n − n

√ = n

+ i:

2 N

!

a a−

=

(a − )an−

a n

G d Tn > N S %P N ' T < ε P ǱL & mH =" 8$ ε L p

n=a

∞ )

(a − )an−

−n

= lim

∞ )

√

an

<

n=b

lim

m−n+ − a − a

#

=w-$x %! G.P

√ n+ n +n−

2 N

!

ak

k=n

=%! G.P

n

na

ka

k=n m )

n

n /n # lim +

L T%! S<

n=

m )

=

k=n

L T%!

n /

yn

< " %+& n→∞ lim

xn

2

!

∞ )

∞ )

a =w-$x n→∞ lim a = 8 d +Q & r4 " " n ^ =an < na G d Tn ≥ N S %P N T^&& 24

n=b

n=a

n=b

! <

xn

n=a

n=b

n=a

∞ )

G.P

∞ )

n

yn

S< < G.P T24 ^ ="& %+& < 4 [Q( " 4 n→∞ lim

∞ ) n=

n

'+ P ! <

@ #1 #:L <@8.a 'I

< <#1 $ Q 8 I

< g(x) ≥ G d Tx ≥ e S mH = g (x) = G d TV+G& x :X L =%! 54 f (x) ^&& − f (e) = f (e) e

T^&& =f (x) ≥ x ≥ e P L & + t $ %&E ^ =%! 54 [e; +∞) @L& & =xn+ = f (n + ) > f (n) = xn G d Tn ≥ > e

f (x)

lim xn

=

n→α

= =

n=

n

= S + x > S = x − x + x − x = S + (x − x ) > S

n→α

= x + x + x + x + · · ·

A7S"

x + x + x + x ≤ x + x + x + x = x = x

== =

n P ǱL & S p f ǱN ! & mH S > S > · · · > S

n

S < S < · · · < S

n+

n+ −

≤ x

n

+ ···+ x n = x ! " n

>S

(n+)

<S

> ··· >

(n+)+

< · · · < S

w>=1 x w9=1 x

%! 4 & ^+H L < Q<_ {S n }αn= @QD mH @QD T

n=

|xn | =

+ ··· + x

i=

= S − x < S

x + x ≤ x + x = x

n+

(− )i xi

= x − x + x

S

x ≤ x

+x

n )

= S + (x − x ) < S

n+ lim (− )n n + lim lim (− )n n→α n n→α lim (− )n

V B

n

= x − x

n→α

="& G.P

x

S

n+ n

+ i: 2W/' 2O ;- C/I= " T24 ^ =%! %De < Q<_ QD {xn }∞ n= 3 ! %! d ∞ x G.P & :

< LO n= n n x

= x >

S

= B< d

∞ )

n=

= S − x < S

N

)n

S

V+P

(− )n xn

" p f =V+S p Sn =

=

L =%! S< − + − + · · · ! S p f =xn = (−

α )

@QD p ^ & =%! ]G.P

V TO& P;! U.B & T^&&

! !

2

TL ="& G.P T%! (Q a >

%! 4 & G.P < Q<_ /na

n

n+

)−

k=

xk ≤

n ) k=

k x

T^&& < %! O& L 2

∞ ) n=

k

≤

∞ ) k=

xn

k x

|xn+ | =

(n + )a

≤

na

= |xn |,

V T( < a) %! 54 f (x) = xa U& $ < k

lim |xn | = lim

n→∞

%De 2f.B & ! mH =%! G.P

C & " B TW&& = / ' L S_& Y+34 C P

@QD TL q%! S<

n→∞

∞ ) n=

na

=

√ (− )n / n n

!

2 N √ n

%fC T ^ @P & =%+ Q<_ |xn | = / n =V+ !& ( ; ∞) @L& & f (x) = x−/x U& M g(x) = ln x −

F6

d 24 S < f (x)

= f (x)

ln x − x

< <#1 $ Q 8 I

n + n + n + n= ∞ )

)

M)( #:L <@8.a ,I

+ (− )n , ) n+ n= ∞ )

∞ )

)

n=

(− )n √ , n + (− )n

∞ )

)

(− )n−

∞ )

)

n=

(− )

n+

n= ∞ )

/n)

n

(− )n

n=

n

,

) )

n=

sin n , n

∞ ) n=

n(ln n) {ln(ln n)}

& 5 T /p−

4

∞ )

*H

! V+hS

n

'+ P ! V+

2

=%! < G.P

∞ ) (− )n+ n

n= n+

∞ ) (− ) n

n=

= xn

n

!

(n )p n

p−

! /< < /< T^&& = < p − =

≤ xn+ =

%! G.P < p

∞ ) n=

!

n ln n

2 N

f (n + )

≤

f (n)

= xn

V+ B TmK! n=

n x

n

=

=

∞ ) n= ∞ )

n

n=

=

ln

Ld && T+ =%! S< ="&

!

'+ P ! Q Tw9=9=1 L w9x %./x %! S< =w 0=6=1 L w x %./x %! G.P

n=

%! G.P /< < /<

<

i &!0 T"D M7; G.P T"& n=a

/np

@QD V+ %&E & {7; ^ & V+ B Tp ^ & =%! Q<_ < %De {xn }∞ n= %! %De [; +∞) @L& & f (x) = x ln x U& M %fC

54 L& ^ & f (x) ^&& < wf (x) = ln x + Lx = T"& Q<_ L& ^ & U& + =%! f (x) x ln x ^&& <

PQ

T"& G.P TM7; G.P ! P %! ^"< =%! s7l {7; ^ m8C Q<

n=

n=

!

|xn |

np

n= ∞ )

=%! S<

!

xn

G.P ! ="& G.P

∞ )

/np

∞ )

n=a

=" +

∞ )

.

< %De 2f.B & ! & UB 3& @.P D/ W(& +/ ^ V+P =& w ! x " , 2f.B & ! " V+N _& @ < & ! G =V& =< G.P ! < M7; G.P !

&!0

∞ )

=

n

n=

J'& !DM 9=7-`

∞ )

≤

(n + )p

=

,

n(ln n) {ln(ln n )}

n x

,

&

∞ ) n=

≤ xn+ =

∞ )

! & 8& ^ >=6=1 Ld ^&& < !P

n= ∞ )

24 ^ =V+S p

∞ ) n+− n− ,

)

2

! < Tp > + i:

(− )n+ , − ( /)n

∞ )

) n sin(

√ n

n=

)

∞ ) (− )n+ , √ n n n=

)

n − , n − n n= ∞ )

)

,

∞ ) n=

L ' P S< < G.P

! +

)

∞ ) n=

FF

nn+ /n , (n + /n)n

)

n ln(n )

n ln ∞ )

n

n= ∞ )

n

n=

n ln n

'+ P ! <

S< ! T >=6=1 !

+ 3& L ! ∞ ) n=

n , n + n

M)( #:L <@8.a ,I

=

lim

n→∞

=

< <#1 $ Q 8 I

n

n+ n

lim

+

n→∞

L T%! M7; G.P

−n n

T"& S< T%! (Q a =

n=

!

n+ lim n→∞ |a|

=

L

+

+ =xn =

+

.. · · · n .. · · · (n + )

=

xn+ lim n→α xn

=

(n + ) lim n→α (n + )

=

lim

'

xn

+ ···

=+ 3&

Q ^ =

n=a

xn

n |xn |

∞ )

! G d T < S w[Q =%! G.P

xn

xn

n

p %Q <

∞ )

n=N

∞ ) n=N

rn = rN

%! "

=%! G.P xn

! G d T > S w

xn+ = V+ i: lim =n→α xn = > w < ≤ ≤ w[Q V+S

A7S"

23(!

%! C r V+ i: = ≤ ≤ V+ i: N C ε = r− ǱL & 24 ^ = < r < x + = n+ − < ε n ≥ N P ǱL & %P xn

|xn | <

n=N

M )

xn+
r

n−N

|xN | = |xN |

∞ )

M−N )

k

r <

k=

n=N

mH =%! G.P

∞ )

rk

k=

+Q & d !P !

< r <

G.P ^&& < O& L _+

∞ )

xn

%De 2f.B & !

n=a

=%!

24

V+ i: ǱL & 24 ^ = > n > N P ǱL & %P N ' ε = ( − )/ xn+ − < ε V xn

x < n+ < xn

−

+

ǱL & < < k 24 ^ k = − " i: S < %+ 4 n→∞ lim |xn | mH =k|xn | < |xn+ | n > N P ∞ )

="& S<

2

xn

! ^&&

n=a

L T%! G.P

n

|xn | <

xn

! G d T < S w[Q

n

∞ )

=<

n+

n=a

! G d T > S w

n=a

23(!

∞ )

=%! S<

M )

V+S p %Q < A7S" r < %P r C 24 ^ ε = r − ǱL & = lim |xn | < r < < r < n→α |xn | − < ε n > N P ǱL & %P N ' + =|xn | < rn mH = ≤ |xn | < r S

x

|xn | < r|xn− | < r |xn− | < · · · < rn−N |xN |

< %! (Q ! 24

n=a

=%! S<

;[& C/I= !

%" T^&&

lim ^ ="& B = n→∞ ∞ )

! + i:

x − r < n+ − < r − xn

2W/' * ) C/I= !

+ < M7; G.P

xn

2 N

+ n +

=> n + n +

n→α

∞ )

+ < M7; G.P

n

+ i:

= ≤ < ]Q < <

24 ^ ="& B n→∞ lim

="& S< " ! mH ∞ )

=wF=6=1 L w0x %./x %! G.P ! '

n=a

. .

2 N

!

n=

=∞>

.

n=

n=a

! G.P

2 N

(n + )!/an+ lim n→∞ n!/an

=

n=

∞ ∞ ) (− )[ln n] ) = n n

= e − <

∞ ) n! an

∞ ) (− )[ln n] n

rn

n=N

FI

=

∞ ) n! nn

n=

!

2

!

(n + )!/(n + )n+ n→∞ n!/nn lim

< <#1 $ Q 8 I

M)( #:L <@8.a ,I

a x cos x→∞ x

x a − = lim + cos x→∞ x

lim x cos a − x = lim ( + u)/u x→∞ =

! ^&& =%! G.P

lim

=%! G.P _+

a x

i: & :X L =u =

lim x

x→∞

cos

a x

−

− cos

a

cos v − = lim v→ v a = −

mH

lim n |xn | =

n→α

= = = =

# lim

n→α

n+

n+ n

n+ lim n→α n lim + n→α

n −

n n

n+ − n −

−

^&&

N

$−

n+ n

−

n n→α n +

lim

) ) ) ) )

)

−

n |xn | − ^&&

≤ lim |xn |

n→∞

∞ )

xn

n=

! ]Q <

|xn+ | |xn |

S " %&E TW&& ! ! && < B _+ lim |xn | G d T"& && < B n→∞ =%! s7l 7 %Q {7; ^ m8C T =%! %Q & DO Ld L ! t $ S T^&&

G d TV+ ! Ld d L V+ ^&& < TV+!& = %Q & TLx V+ ! V+ . _+ " @ Ld L =w+ P = n→∞

n

$ =+G& p

∞ ) an na

2

!

!

lim

n→∞

=

Q T%! G.P

" !

an = lim n a n→∞ n =

+ + + · · · + ! ! n! (!) (!) (n!) ( !) + + + ···+ + ··· ! ! ! (n)! × ! + × ! + × ! + · · · + n n! + · · · n × ! × ! × ! n n! + + + ···+ n + ··· n ( !) (!) (!) (n!) + + + · · · + n + · · ·

+ × × × × + + ··· ×× + × + × × × ×× × × × + ··· + ×× ×

N

n

+

n=b

+ 3& L n + · · ·

<

="& G.P .

2

=%! G.P " ! mH

)

<

! L ' P S< < G.P

n |xn |

<

n→∞

(e − )− × e−

∞ = lim

lim

S p f T%Q ^

+

<

a

<

24

n

=%! . P& < = e−a /

|xn |

n=N

−

x

α )

ǱL& 24 ^ = > V+ i: Tε = = |xn | − < ε n > N P ǱL & %P

u→∞

V Tv =

+Q & d !

< r <

|a|

∞ ) an na

a

|a| √ ( n n)a

= |a|

! G d T|a| < S T^&&

n=b

="& S< ! ^ G d T|a| > S

Ta = − S < %! S< ! n& G d Ta = S n T^&& =%! S< ! + < a a = n(− )n G d /< < /<

∞ ) an na

n

! V+hG& V+ .

n=b

G.P T

= =

F1

= a ∈ R

=|a| < %! G.P

∞ )

cos

n=

a n n

a n n lim cos n n→∞

a n cos n→∞ n lim

!

2 N

L T"&

#5*#N; 9.a %& 2I

< <#1 $ Q 8 I

∞

-

dx x +

=

lim

a→∞

=

a

a→∞

" i: S L T%! G.P

n e

n=

dx

x +

lim arctan a =

∞ )

)

√ − n

n=

)

π

!

2 N

)

√ − x

< %! %De [; ∞) @L& & y = f (x) G d Tf (x) = x e √ √ [; ∞) @L& & y = f (x) mH Tf (x) = x( − x)e− x $ √ V u = x i: & T^+t.P =%! Q<_ -

∞

x e−

√ x

dx

∞

= = = ()

= =

= ( )

=

=

-

−

)

P () e

)

an ,

n=

)

∞ ) n=

)

∞ ) n=

n(ln(n))p

)

,

h_B PC. @QD S 2f.B & QD

∞

{yn }n=b

n e

√ − n

n−ln(n)

),

n n +

√

−

n

,

∞ ) nln(n) , (ln(n))n n= ∞ ) cosh πn . ) ln cos πn

)

,

−

e

n=

n= ∞ )

∞ )

,

n=

n=

1H

! &/ ! L Zj(& P ! Ld ^ @.P +& 8 fk.C T+ ="& +& T P =V& d L & T%+.P {+ & =%+ d 5& y = f (x) + i: E- C/I= " < LO " T24 ^ ="& (; ∞) & Q<_ < ,G %! d "& G.P

∞ ) n=

)

na

∞ )

n n + n − n= ∞ )

xn

-

! G d T + 4 & 8 8" & "& %De =%! G.P

n=c

xn yn

k+

=

f (x) dx k

-

k+

f (k) dx = f (k)

< k

V T[7 ( P n ǱL & ;& ^ U.B & -

n

f (k) <

f (x) dx <

k=a+

a

n )

f (k)

k=a

" + Tn → α ǱL & ^+:X L +S & < ∞ )

f (n) ≤

n=a+

!

k+

f (k + ) dx k

.

n=a

∞ )

P L& mH T%! Q<_ f $ A7S" + f (k + ) ≤ f (x) ≤ f (k) V

f (k + ) =

,

:Q )& C/I= "

< "&

∈ [k; k + ]

n+ )

n ln(n) ln(ln(n))

f (n) ! 8 & :

- ∞ ="& G.P f (x)dx a

,

∞ )

∞ )

n=a

x

n=

, arctan n

∞ )

,

∞

TP(Q N+N C p < a + i: " 3& " ! L ' P G.P 24 ^ +

)

n=

+ cos n + cos n

n

n=

!4 ("!K E 8-` $

< %! 6 @B L 7.B $ ' P (u) U& w x

" ! ^&& =%! " ! , +HP @+A/ L w0x ="& G.P

∞ )

∞ )

√ ∞ ) n ( + (− )n )n

)

, n ln(n)

u e−u u du

u e−u du - a u e−u du lim a→∞

a −u lim P (u)e a→∞ −a − lim P (a)e − P ()e a→∞ P (a) P () lim − a→∞ ea e () P (a) P () lim − a→∞ ea e

∞ )

∞

f (x) dx ≤

a

=%! . P& < ∞ ) n=

n +

%De < Q<_ [; ∞) & y FJ

f (x)

n=a

2

S L q%! G.P

∞ )

= f (x)

2

"

G d Tf (x)

=

x +

< <#1 $ Q 8 I

#5*#N; 9.a %& 2I

∞ ) (− )n n ) + (− ) n

{ < T%! | ^ @74 L V8 ^ 2DE =S 2DE _+Q d

n=

) ) )

+

p

−

+

−

p

+

p

−

+

+

∞ )

G.P ! '

−

−

−

p

+

+

xn

−

+

−

p

+

+

+ i:

− ···

' Ta ∈ R

+ · · · (p > )

p +

+

−

n=

|Sk |

n=a

! T24 ^ ="& 8 < G.P @QD ' {yn }∞ n=b < ∞ )

2

"

n

7G= C/I= " "

=%! G.P

!

< xn = sin(ax) i: & L T%! G.P ! 4 & G.P < Q<_ T%De {yn }∞ @ QD 8 & B & < n= < %! yn =

− ···

∞ ) sin(an) n

= ()

=

xn yn

k k ) ) xk = sin(na) n= n= sin (k + ) a sin k a sin a

n=c

24 ^ =Zn = y − yn < y = n→α lim yn V+ i: A7S" & & mH =%! 54 +l {Zn} < n→α lim Zn = < Zn ≤ xn yn = yxn −xn Zn

=%! G.P

α )

xn Zn

n=a

α )

xn yn = b

n=a

α )

xn −

n=a

α )

78 Ld +

≤

! V+S + 78 Ld L TmH q%!

="& G.P w%! " ! .P x

xn Zn

< xn

=

π (a)n

2

S i: & L T%! G.P

n=

an

V yn = sin ∞ )

xn

∞ )

=

n=

n=

lim yn

π

n

/

=

=

<

x cos x − sin x x x − tan x ≥ x cos x

π/n+ < π/n

yn+ = f

n+

=

n

n −

−

+ ··· n

V+ i: =+G& p S [5

x = x = x = x = · · · = xn+ = xn+ =

< %! 4 & G.P < %De TQ<_ yn @QD T24 ^

;

=%! 54

π

+

π

|Sn | =

=

G d Tf (x) = sin x/x S L T%! 8 yn < f (x)

n −

x = x = · · · xn = −

sin x = lim x→ x

=

+ + − + ···

< 24 & xn < yn

n

π (a)n

n→∞

−

n

π = a − − a π π / n lim sin n

=

n→∞

π

π a

=

+

"

sin

2 N

!

···+ ∞ )

xn yn

w=%! " 2DE >=F= ,e L w>x %./ w x <x

n=a

>

∞ )

n=a

$ %.! ! mH TG.P %! %.! ! < $

2 ="& G.P _+ ! G d T|a|

a sin

π

n

π

f (x)

n ) xk k= ⎧ n = m + ⎨ n = m + ⎩ n = m

p ! .P x & + < +

∞ ) n=

xn yn

= yn

∞ ) cos(an) n

n=

I)

≤

! T9=I=1 M&; mH

Ld '. & & L ! G.P )

S S S

="& G.P w%! ! "

+P 78

#5*#N; 9.a %& 2I

< <#1 $ Q 8 I

ya+ y y + a+ a+ + · · · ya ya+ ya y yn yn− · · · a+ ··· + yn− yn− ya y y yn = xa + a+ + a+ + · · · + ya ya ya xa = (ya + ya+ + · · · + yn ) ya xa tn = ya

< xa

G.P

+

< xn T

∞ ) n=

=

<

∞ )

xn

=

a n+

mH =

b

yn

=

lim Sn

=

C L QD

G d T < S =

{xn }∞ n=

S

= lim n n→∞

xn − xn+

< LO " 24 ^ =yn

=

:X L =

yn

%! d

∞ )

/n

xn

n=

V+ i:

!

∞ ) n− n+

)

A7S"

)

! G.P & :

) 5

<

=

(n + ) nk k + n k(k − ) k + + + ··· n n

∞ ) n= ∞ ) n=

∞ )

xn

G.P _+

/

a n

xn yn

b

2

−

=−

&& p ! T^&&

n−/

)

∞ ) n+ n!

n=

p(p + ) · · · (p + n − ) × p n! n n n + n

S

1$ . & C/I=d # "

∞ )

xn

G d T"& G.P

n=a

∞ ) n=a

∞ )

yn

n=a

yn

<

y xn+ < n+ xn yn

=%!

V+ i:

> n

n=

mH Ta < b

n ≤ a P ǱL & _+ < "& %De 2f.B & !

k k(k − ) + + ··· n n k(k − ) + ··· = k+ n xn L S_& k n→∞ lim n − > S ^&& xn+ ∞ ) < Ld & & mH T%! S< yn ! $ < %:S ' xn − xn+

a < b

−

+ n

! mH

n=a

k

=

)

n=

yn xn > xn+ yn+

>

^&& <

b

L ' P G.P T&d Ld '. & " +P L !

< & %De ∞ )

n→∞

=%! G.P T"&

n=a

xn xn+

b

n=

b a n+ ÷ − = yn+ b

n→∞

=%! S< ! T > S < %! G.P k

a n

a n

lim yn = lim

G) C/I= "

xn =

n=

^+t.P =%! 54 yn @QD T5

=%! . P& <

2

<

÷

<

xa lim tn ya n→∞ ∞ xa ) yn ya n=a

≥

∞ )

$ T
n→∞

n=a

a n = / − b ∞ n )

$ T24 ^ =yn

<

^&&

2 N

S

< a < b

V+ i: T{7; ^ 2DE & =%!

b

b

! G d T

b n − an n

Sn

=

xa + xa+ + · · · + xn

tn

=

ya + ya+ + · · · + yn

A7S"

24 ^

Sn

)

=%! S< _+ xn ! TN w^.x =S 2DE & 24 & < %Q I

x = xa + a+ + xa xa+ = xa + + xa xn xn− ···+ xn− xn−

xa+ xn + ···+ xa xa xa+ xa+ + ··· xa xa x · · · a+ xa

< <#1 $ Q 8 I

xn =

× · · · (n − ) × · · · (n)

- . / 0*1 II

p

= = = =

P

=

P ǱL & S 12 ) E C/I=d " xn < S G d T = lim n ln < xn > n ≤ a n→

24 ^ =+G& p ^&& <

=%! S<

xn lim n − n→∞ xn+ p n + lim n n→∞ n + − − x lim ( + x)p − x→∞ x ( + x)p − lim x→∞ x p( + x)p− p lim =

x→∞

∞ )

! > S < G.P /nk

∞ )

=S 2DE & 24 & <

2 " &!'P0U n @7.B { x

∞ ) α n ( + a) = a n n=

T5 q"& a = − L &

xn

=

< α − n + G d Tn > n := [α] + S 24 ^ V+ = P %fC VP P an @.P 5& & n L ^&& < 24 ^ = De d @.P V+ i:

%! L 8" & ! ' N @D!3 K+

sum(x(n), n = a..infinity) −−−−→

n=

sum((−1)∧n/n∧ 2, n = 1..infinity) −−−−→ −

cd

xn = lim n − n→∞ xn+ n+ = lim n − n→∞ n−α n(α + ) = lim =α+ n→∞ n − α

! α < /< T^&&

! .

∞ )

! @D!3 . & K+

α (− )n n α(α − ) · · · (α − n + ) (− )n n!

:=

(&4+ ( ^ )4 ; A

V

"

α

k T D!3 K+ t $ P fC inﬁnity T" %+& B S =. P fC ="

∞ ) (− )n n

S< T{+ ^

yn

n

limit((1 − 1/n) (2 ∗ n), n = infinity) −−−−→ e

x(n)

A7S"

n=a

< %! N+N C a + i:

%! L 8" & QD ' @D!3 K+ limit(x(n), n = infinity) −−−−→ x(n) @QD n V n→∞ lim ( − /n) @D!3 . & K+ −2 ∧

L

∞ )

S< &

n=a

2

HH

xn

(&4+ ;-& ^ 43 ; A

n=a

V+ i: _+

!

& < q ( ' L S_& k > S T^&&

%3 W(& & TK+ _: L ! 2N @P & =" 5B ' j: L ^+.P

5& &

xn

xn yn xn > ⇔ > + k xn+ yn+ xn+ n xn + > k ln ⇔ ln xn+ n xn − ⇔ n ln > nk + − · · · xn+ n n n xn k k ⇔ n ln + − ··· >k− xn+ n n

+, - ./(0

∞ ) n=a

24 ^ =yn =

%Q =

7 24

xn

n=a

! G d Tp < S T+ =x = n + ="& S< ! T < p S < %! G.P "

7 24

xn+

n=

∞ ) α(α − ) · · · (α − n + ) (− )n xn = n! n=

=%! G.P

2

π 12

!

http://webpages.iust.ac.ir/m_nadjafikhah/r1.html

=%! "
I0

p +

× ×

p +

×× ××

p

+ ···

2 N

! & '()"

< + Td UD & < _+Q d ^ . L 8 U& L F + i: =%! {N @ 7 T,G [5 S @ C. & w7.B $ x "& ! d =%! Df ⊆ S @ & 5& y = f (x) < " V+ C V+ & S & %: fn (x) ∈ F U& & DN 8" & / < -D & f (x) =

∞ )

fn (x)

n=

& &B ^ : j: ^ L vP -. n; C ^ %H V+P ,! ^ &

;N G.P =J 8"

∞

2

5& @QD =+&+& QD ^ ;N =+G& p = ^ : & =D = R < fn (x) = +x n V+ .C L \< & d ;N +x

n

n=a

n→∞

n→∞

+x

n

=f (x) = ^&& < n→∞ lim x n = ∞ G d T|x| > S = ^&& < lim x n =

G d T|x| = S n→∞

lim x ^&& < n→∞

n

=

G d T|x|

<

S =

< C = R . mH =f (x) = ⎧ ⎨ f (x) =

Q< !+H P

⎩

fn (x)

/

|x| > |x| = |x| <

S S S

. " p f

+

=

+

=

I

=V+ Lld [5 $ & P L & + i: { @ ="& D @C. & U& ' fn (x) D fn (x) n =V+ ∞ ' {fn (x )}n=a x ∈ D P L & %! \< & 5& C & 5& @QD P 5 T%! C @QD =. j yQD z N < N+N +a G.P {fn(x)}∞ @ .P C @C. n=a P x ∈ D x ∈ P L & S =V+ %! lim fn (x) = f (x) C f (x) U& Tn→∞ V+ < + {fn (x)}∞ n=a n

≥ a

{fn (x)}∞ n=a

'8C!

;N

f (x) −−−−−−→ lim fn (x) = lim

f (x)

34 O7

2F! '8P0U d 0 '8P0U '8()"

'()" !0 '8C!

&!'QK"

;N

`M@!

lim fn (x) −−−−−−→ f (x),

n→∞

%+ f (x)

C

&

=" B =J 8" & I>

45 <#1 $ Q 8 J

< " [5

U

&

45 Q 8 J

<

g(x)

f (x)

) f U ≥

)

f U = ⇒ f ≡ U

)

af U = |a| f U

)

f + gU ≤ f U + gU

&

S

G d Ta ∈ R

≤ |f (x)| ≤ sup{|f (x)||x ∈ U } = =f (x) = ^&& < V+ B w>x 2DE & =

sup{|af (x)| | x ∈ U }

=

|a| sup{|f( x)| | x ∈ U }

=

|a| f U

|a + b| ≤ |a| + $ S

f + gU

C [5 L T
sup{|f (x) + g(x)||x ∈ U }

≤

sup{|f (x)| + |g(x)||x ∈ U }

≤

sup{|f (x)| | x ∈ U } + sup{|g(x)| | x ∈ U }

=

f U + gU

=

lim fn (x) x n = lim + n→∞ n x x n/x = lim + = ex n→∞ n n→∞

. T^&& =f () = fn () = lim

n→∞

+

x n ;N x −−−−−−→ e , n

G d Tx = S <
R &

5& @QD + i: 7Q*Q 43 4 =%! ;N G.P f (x) & C & {fn (x)}∞ n=a ∞ f (x) & U ⊆ C & {fn (x)}n=a 5& @QD V+hS S %: Nε (Q ε > P L & %! V+"& " (Q n ≥ Nε P < (Q x ∈ U P L &

V+ %Q ^ =|fn (x) − f (x)| < ε

&!0

D. .

88

p f w9x ^+t.P mH T|b|

=

2 N

x n ∞ 5& @QD n n= n =D = R < fn (x) = + x n

G d Tx = S " + e +

=%! & w x V8 |f (x)| ≤ x P ǱL & $ A7S" & G d U & f ≡ S V+ B w0x V8 2DE & f U = ]Q < |f (x)| = x ∈ U P ǱL & %! ǱL & mH =sup{|f (x)| | x ∈ U } G d f U = S m85Q& x ∈ U P

af U

=+G& p

lim fn (x) −−−−−−→ f (x),

n→∞

U

&

=" B 0=J 8" &

=%! . P& <

2

5& @QD 8 & : < LO " ! %! d "& 88 G.P y = f (x) & U & {fn(x)}∞ n=a P L & S %: Nε ' Tε > P L &

G +& & =fn(x) − f (x)U < ε n > Nε 88

lim fn (x) −−−−−−→ f (x) ,

n→∞

U

&

⇔ lim fn (x) − f (x)U = n→∞

& ^&& =n→α lim fn (x) − f (x)U = V+ i: A7S"

G d n ≥ N S %P N ' ε > P ǱL ,5 X & Tfn(x) − f (x)U < ε sup{|fn (x) − f (x)| | x ∈ U } < ε

U

& 88 G.P 0=J 8"

[5 S ⊆ R & y = f (x) + i: V+ [5 L 2j& S & f (x) =" =f S := sup {|f (x)| |x ∈ S }

D. . F"

e 5 & H! Supremum @.7 [( sup & G d T"& !+H S & f (x) S %! / =%! =S ! w"& Maximum [( x max L sup d O& ^ 8$ 5 A @C. H! TM+/ ;& =sup[; ] = sup[; ) = fke =C. I9

45 Q 8 J

= =

=

45 <#1 $ Q 8 J

lim sup { |fn (x) − f (x)| | ≤ x ≤ } lim sup − ≤ x < n→∞ +x n ∪ − x = # $ x n lim sup ≤ x < n→∞ +x n

V sup [5 M&; T^&&

n→∞

()

= w%: 8& w x ,e t d .P "< &x w x < =+ 2DE 5& QD G d TU ⊆ [; ∞) " i: S , ^ & ="& 88 G.P 4 & U & i<

∀ x ∈ U : |fn (x) − f (x)| < ε

=%! G.P U & f (x) & 8" ' X & {fn(x)} @QD mH & 8" ' X & U & {fn(x)} @QD V+ i: , & %P N (Q ε > ǱL & mH ="& G.P f (x) + =|fn(x) − f (x)| < ε n ∈ U P n > N P ǱL

=

& T24 ^

sin(nx) =fn (x) = n

|f (x)|

= = ≤

2 N

+ i: x ∈ R P L

{|fn (x) − f (x)| | x ∈ U } < ε

V Tsup [5 && mH fn (x) − f (x)U = sup{|fn (x) − f (x)| | x ∈ U } < ε

U = [; /]

lim fn (x) n→∞ sin(nx) lim n→∞ n lim

n→∞

n

lim x − U

n→∞

=

n→∞

=

^&& ;N

2

& {xn }∞ " @ QD n= L =%! f (x) = & 88 G.P

n

=

lim fn (x) −−−−−−→ ,

=n→α lim fn (x) − f (x)U = + <

2

R &

lim sup xn ≤ x ≤ n→∞ n lim = n→∞

;N

L T%! 88 G.P ^ T
f (x) −−−−−−→ lim fn (x)

=

n→∞

=

= [; ]

& Q L q%+

lim xn

n→∞

lim fn (x) − f (x)R =

≤x< x=

n→∞

= = =

S S

lim sup { |fn (x) − || x ∈ R} | sin(nx)| x ∈ R = lim sup n→∞ n

=

;N U& & 88 G.P U

n→∞

lim

n→∞

lim

n→∞

lim

n→∞

n n n

lim xn − f (x)U = lim sup { |xn − f (x)|| ≤ x ≤ }

n→∞

4 P x

sup { | sin(nx)|| x ∈ R} sup { | sin x|| ≤ x ≤ π} =

lim fn (x)

= ()

=

( )

=

P

=

=

L & y = f (x) < | n − f ( )| = $ V+& V+ O& 2DC & ^&& T%! =

2 N

=U = [; ] < fn (x) = nx( − x)n + i: S =fn(x) = G d Tx = x = S T24 ^ ^&& < < − x < G d Tx ∈ (; ) n→∞

n→∞

= =

88

y = f (x)

lim nx( − x)n

lim sup { xn | ≤ x < } n lim lim x

n→∞

n→∞

n→−

=

lim

n→∞

& {fn(x)}∞ @ QD G.P T+ n= =%+

n→∞

n an y x lim y y→∞ a x lim

88 G.P [;

n→∞

x lim

y→∞

ay ln a

=

lim fn (x) − f (x)U

n→∞

I6

∞

2 N

& @QD + x n n= L =w0= =J ,e L w x %./x %+ ]

45 <#1 $ Q 8 J

45 Q 8 J

G.P +Q & 24 ^ =x ∈ U V+ i: B^ 7S" N ' ε > ǱL & f (x ) & x = x {fn} ;N =fn (x) − fm (x ) < / n, m ≥ N P ǱL & %P ǱL & ^&& T%! G.P U & 8" ' X & {fn } $ < m, n > N P ǱL & %P N ' ε > P S , =fn (x) − fm (x ) < ε/(b − a) x ∈ [a, b] P

G d x ∈ [a, b] < m, n > N < N = max{N, N }

L w0x < a =

%! " i: w x 8 Y+ ^&& =%! " ! =>=1 ;N lim fn (x) −−−−−−→ , [; ] & n→∞

G d Txn =

lim fn (xn )

^&& fn (x) − fm (x) − fn (t) + fm (t) = = (x − t)(fn (β) − fm (β)) ε ε < x − t ≤ (b − a)

< x ∈ [a, b] P ǱL & T =n, m ≥ N < x, t ∈ [a; b] S V n, m ≥ N P fn (x) − fm (x) − fn (x )

+fm (x ) − fn (x ) + fm (x )

fn (x) − fm (x) − fn (x ) + fm (x ) +fn (x ) − fm (x ) ε ε + =ε

<

=%! 8" ' G.P [a, b] & {fn} ^&& < G.P g & {fn } < !+H .P P fn i: M&; $ V y ∈ [a; b] P ǱL & =%! 8" ' lim

y

fn (x) dx =

-

y

g(x) dx - y lim {fn (y) − fn (a)} = g(x) dx

n→α

n→α

a

a

a

U

y

n→∞

−

lim

x→∞

n+ n+ x x

= e −

& {fn (x)}∞ @ QD + i: n=a 5& =%! 88 G.P y = f (x) &

_+ f (x) G d T"& !+H x ∈ U fn (x) %D/C S w[Q

x→x

n→∞

lim fn (x)

x→x

< & ;N G.P f (x) & [a; b] ⊆ U & fn (x) S w n P L & < "& 88 G.P [a; b] & {f (x)} 88 G.P [a; b] & f G d T"& !+H [a; b] & fn =n→∞ lim fn = f < %! & {fn} < & ]KQG [a; b] & fn n _P L & S w| [a; b] & _+ f (x) G d T"& 88 QG.P f & [a; b]

b

f (x)dx = lim

n→∞

fn (x) dx

n P ǱL & $ =x ∈ U V+ i: B_!" A7S" δ > ' ε > P ǱL & mH T%! !+H x

G d Tx ∈ U < |x − x | < δ S %P a

a

fn (x)

fn (x) − fn (x ) < ε

N ' ε

> ǱL & mH f (x ) = lim fn (x ) n→α fn (x) − fn (x ) < ε/ n ≥ N P ǱL &

$ %P ε > ǱL & mH T%! f & 8" ' G.P {fn } $ x ∈ U P < n ≥ N P ǱL & %P N ' Tn ≥ N = max{N , N } S mH =fn (x) − fn (x ) < ε/ V x ∈ U < |x − x | < δ f (x) − f (x )

& {fn} Q< + n→α lim fn (a) = f (a) <

-

−

lim

^&& =%! " ! =>=1 @+A/ L w>x

=%! 4 [Q( + < n→∞ lim ||fn (xn ) − ||U ≥ e−

lim fn (y) = f (y) c< T%! ;N G.P f

n→α

= =

ϕ(x) − ϕ(t) = (x − t)ϕ (β)

≤

S L T%+ 88 G.P ^

()

@+A/ & & 24 ^ =[t, x] ⊆ [a, b] < ϕ = fn − fm V+S < t < β < x %P β R SO

=

n+

n→∞

fn (x ) − fm (x ) < ε , ε fn (x) − fm (x) < (b − a)

fn (x) − fm (x)

−x

=

≤ g(x) dx = f (y) − f (a) =⇒ f (y) = g(y)

f (x) − fn (x) + fn (x)

−fn (x ) + fn (x ) − f (x )

f (x) − fn (x)| + |fn (x) − fn (x ) ε +fn (x ) − f (x ) < = ε

a

IF

45 Q 8 J

45 <#1 $ Q 8 J

V x & x 5 & < − x + x − · · · + (− )n x

88

−−−−−−→

V U = [−a; a] ⊆ (−

n

+ ···

x +

; ) &

,

[−a; a] &

=[a, b] & n→α lim f (x) = f (x) mH P ǱL & %P N C ε = ǱL & B` 7S" + =fn (x) − f (x) < x ∈ [a, b] P < n ≥ N = fn (x) + f (x) − fn (x) ≤ fn (x) + f (x) − fn (x) < fn (x) +

f (x)

T+GQG < +GN &

+ x + x + · · · + (n + )xn + · · ·

88

−−−−−−→

( − x)

− x + x − · · · + (− )n (n + )xn + · · ·

88

−−−−−−→ x−

x−

x

x

+

+

x

− · · · + (− )n

n+

x n+

( + x)

+ ···

88

x

+ =%! [a, b] & mH T%! ]H ,G fn $ %P N ' (Q ε > ǱL & =%! _+ f & =fn (x) − f (x) < ε G d Tx ∈ [a; b] < n > N S

b−a {+ ^

−−−−−−→ ln(x + ) − · · · + (− )n

x n+ + ··· n +

-

V+ i: 1?/' C/I=d # & U & {fn } 8 & : < LO " =%! U & QD ε > P ǱL & %! ^ "& 8" ' G.P f 5& x ∈ U P < n, m > N P ǱL & %P N ' =|fm(x) − fn (x)| < ε

8" ' G.P f 5& & U & {fn } V+ i: A7S" n ≥ N P ǱL & %P N ' ε > ǱL & mH ="& P ǱL & =|fn (x) − f (x)| > ε/ x ∈ U P < V x ∈ U P < n, m ≥ N |fm (x) − fn (x)|

=

|fm (x) − f (x) − fn (x) + f (x)|

≤

|fm (x) − f (x)| + |fn (x) − f (x)| ε ε + =ε

<

ǱL & %P N ' ε > P ǱL & V+ i: , ǱL & =|fm(x) − fn(x)| < ε x ∈ U P < m, n ≥ N P < r4 " " {fn(x)} @QD T%&E x ∈ U P @QD V+P =%! G.P f (x) C & ]Q =%! 8" ' G.P x → f (x) @;& & f U& & fn " α & n + < m ≥ N ^ :S %&E & V+S + |fm (x) − fn (x)| < ε

ε dx = ε b−a

a

2 {fn }

a b

≤

88

−−−−−−→ arctan x

=%! G.P

-

b

f (x) dx a

&

x n

-

b

fn (x) dx a

@QD ^&&

2

=ex := n→∞ lim + [5 M&; n @QD [a; b] @ & @L& P & + %&E ^. C& x n x $ T+ =%! 88 G.P e U& & + n

(ex )

p

=

n=

k=

" p f =+G&

−x

n→∞

TU

2 N

n=

;N

lim fn (x) −−−−−−→

,

(− ; ) &

S +P ^. C & "& 88 2j& U & G.P G d

= [−a; a] ⊆ (− ; )

+ x + x + · · · + xn + · · ·

88

−−−−−−→

−x

,

[−a; a] &

V −x & x 5 & T^&&

fm (x) − lim f (x) < ε

− x + x − · · · + (− )n xn + · · ·

n→∞

2

b = {fn (x) − f (x)} dx a - b fn (x) − f (x) dx ≤

- b b fn (x) dx − f (x) dx a a

88

=%! . P& < =|fm(x) − f (x)| < ε

−−−−−−→

II

x+

,

[−a; a] &

45 <#1 $ Q 8 J

Sa+ (x)

=

Sn (x)

=

45 <#1 J

G.P < ;N G.P !& " PC. & " PQD 88 +

fa (x) + fa+ (x), · · · n ) fk (x), k=a

) fn (x) = xn − xn+ ,

. & < + {fn(x)}∞ n=a

)

fn (x) = xn − xn+ ,

U = [; ],

U = ; ,

{Q; @.P mH T%! 5& @QD ' 5& ! P $ L = +H s& _+ Zj(& @ ! ^ & D/ W(& =88 G.P 7.B

)

fn (x) = x − x

U = [; ],

)

fn (x) =

U = (; +∞),

fn (x) =

U = [; ],

U = ; ,

Q @QD & 5& ! QD ^ =V+P

∞ )

fn (x)

=V+ [5

n=a

C

&

∞ )

5& ! + i:

f (x)

)

& : < LO " =%! ;N G.P y = f (x) & < ε P L & %! d 5& ! ^ 88 G.P n > Nε P L & S %: Nε '

fk (x) − f (x)U < ε

k=a

/< < /< TG +& & 88

∞ )

fn −−−−−−→ f (x),

U &

U

="& F= =J @+A/ L j:f& + ^

2

G.P y = f (x) & U &

∞ )

fn (x)

S

n=a

G d T"& ]H,G [α; β] ⊆ U & P fn (x) @.P S w|

β

f (x) dx = α

2

∞ ) n=a

)

fn (x) =

x , + xn

U = [; ],

)

fn (x) =

nx , +n x

U = [; ],

)

fn (x) =

nx

U = ( ; +∞),

)

,

+n x

fn (x) =

x +

n

U = R,

,

sin(nx) , n

U = R, U = (; +∞),

fn (x) = arctan(nx),

)

fn (x) = n(x n − ), U = [ ; a], < a,

∞ +P G.P [; ] & −nxe−nx

5& ! ;N

n=a

G d T"& 88 _+ f (x) G d T"& !+H x ∈ U P fn (x) @.P S w[Q =%! !+H x _+ f (x) G d T"& !+H x ∈ U P fn(x) @.P S w ∞ ) =f (x ) = fn (x )
xn , + xn

) fn (x) =

n=a

k=a

,

n

)

6 n 6 6) 6 6 6 =n→∞ lim 6 fk (x) − f (x)6 =

6 6

n

, x+n ) fn (x) = nx , +n+x

n=a

n )

n

=n→∞ lim

-

fn (x) dx =

f (x) -

n=

S T
G d T"& "

w 6

f (x) dx

+P w F

- n + sin x dx = n→∞ n + cos x lim

5& @QD 88 G.P L .Px w=" ! [; ] @C. & ,G

β

fn (x) dx

34 9!0

α

="& 6= =J @+A/ L j:f& + ^

T%! C ! ' ;N P 5& ! P $ & C ! G.P Ld . mH 8" & = V+.5 5& ! & ;N 24 -;$ +S:

I

! =VLH C ! V+.5 & W(& ^ =%! 5& PQD L Zj(& ' 5& 5& @QD ' {fn(x)}∞ n=a + i: 24 & B 5& @QD ="& D @ &

Sa (x)

I1

=

fa (x),

45 <#1 J

45 <#1 $ Q 8 J ∞ ) xn = f (x) n!

=

! 8 & : < LO "

n=

%! d "& 88 G.P U

+ < df (x) = dx T df (x) = f (x) T^&& f (x) dx =f (x) = Aex T^&& =%! %&E C A Tx + ln A =f (x) = ex + =A = Ae = mH Tf () = " %&E 5 ln(f (x)) =

88

ex −−−−−−→

p U

+x+

= [; a]

x

&

m ) x e−kx

xn + ··· [a; b] & + ··· + n!

∞ )

x e−nx

n=

k=n

k=n m )

≤

2 N

24 ^ =+G&

m )

=

5& !

m ) = fk (x) < ε m 6)

=666

k=n

N>

ln

a

k=n

∞ )

f (x) :=

m ) xk k!

∞ )

=

k=n

x

n

C n!

<

Cn n!

∞ )

(− )n

n= ∞ )

Ck k!

k=n m )

C C C × × ···× n+ n+ k k=n m k−n ) k=n

" i: S mH

G.P

U

&

ε(C)n log +
n= n

∞ ) x =f (x) = n! n=

!

k=n

f (x)

=

+

∞ ) xn n!

n=

x n . (− )n (n)!

=

n=

∞ ) nxn− n!

n=

IJ

< ε

G d

U

V+"& " U & V+ i: =%! 88

x n+ , (n + )!

m k (C)n ) n! k=n k m (C)n ) n! k=n n− (C)n n!

<

(e−x )n

V+ [5 + i:

:=

=

=

+ <

x − e−x

=

2

m Cn ) C k−n n! (n + )(n + ) · · · k

=

n=

C(x)

k=n m )

x e−nx

∞ )

P < n ≥ Nε P L

m ) |x|k k!

≤

=

n=

≥ Nε

i: & G d TU = [a; b] ⊆ R S V m ≥ N < n ≥ N TN = ([c] + )

≤

G d

x e−nx

5& !

n=

k=n

n=

S(x) :=

∞ ) xn n!

U

= max {|a|, |b|}

6 6 m 6) 6 6 −kx 6 x e 6 6 <ε 6 6

≥ Nε

U

=+G& p

− (e−a )m−n+ − e−a

a ε( − e−a )

P L &

5& PQD & " @+A/ L j:f& + ^ 2 ="&

S mH

< ε

<m

6 6 fk (x)6 < ε m ≥ Nε 6

a (e−a )n < ε − e−a

<

5&

& S %: 6 Nε ' < ε P L & TG +& & 6

a e−ka

a e−na

∈ U

fn (x)

n=a

k=n

TC

x e−nx

V+"& " x

∞ )

&

= [a; b]

P L &

S %: Nε '

k=n

=

2W/' C/I=

24 ^

45 <#1 $ Q 8 J

G.P R & =fn (x) ≤

n

∞ ) n=

ANA #:L <@8.a J

G.P 5& !

x +n

G d Tfn(x) =

=%! G.P wa

=

2 N

S L T%! 88

x +n ∞ )

& !x

n

n=

C !

G.P U & S G d TU = [a; b] ⊆ R S +P

5& ! 88 G.P c "<−M Ld && mH =V R & p & @L& T88 G.P @ " B %+ & L 5& ! 88 G.P +P c "<−M Ld '.

∞ ) (− )n ) , x + n

) ) )

∞ )

nx , + n x

n= ∞ )

arctan

n=

x

,

x + n

(x + n)(x + n + )

n=

U = [; ]

&

&

∞ )

fn (x)

(

n=

5& ! S

− x)x

U

2%

&

& < M

! $

A7S"

ε > ǱL & mH = m ) ^ & = xi < ε m, n ≥ N m ) i=n m ) i=n m )

fi U sup{|fi (x)| | x ∈ U } xi < ε

n=

xn+ xn − n n+

≤x≤

& " " {fn} mH =%! 8" ' G.P

U

5& !

2

S L T%! 88 G.P U = [− ; ] n xn+

G d Tfn (x) = x − n

fn (x) =

7G= C/I=

⇔

xn− − xn =

⇔

x=, x=

i: & mH =U & fn (x) ≤ xn =

n

−

n

−

n+

n+

^&&

n+

8 & B & <

=c ≥ max{a, b}

∞ )

xn

=

n=

@QD > %De C ' L & V+hS U & =|fn (x)| < M x ∈ U P L ∞ {fn (x)}n=a

∞ )

<−

n=a

n=c

'

& ^&& < r4

2

+P

xn

n=a

i=n

88 U & {gn (x)}∞ @ QD < "& 88 G.P n=b 8 {gn (x)}∞ @ QD x ∈ U P L & < "& n=b C ∞ ) T%! 88 G.P fn (x)gn (x) 5& ! G d T"&

!"! 4

U

<

. c "< Ld L ^ Q< T%! 88 =%:S + U

≤ =

; ,

∞ )

i=n

n

=%! 88 G.P _+

N

V T{+

i=n

24 ^ T|fn (x)| ≤ xn

fn (x)

n=a

P ǱL & %P

U = R. ∞ )

∞ )

< M7; G.P U &

U = R.

U=

C/I=

x ∈ U P L & < "& G.P < %De 2f.B & ! '

6 6m 6) 6 6 6 fi 6 6 6 6

−M

PW

, U = (; +∞),

∞ ) sin((n + )x) , n(n + )

G.P

+ i:

∞ ) n √ xn , ) n! n=

)

xn

I

n=a

U = [; +∞),

∞ ) n=

∞ )

r4 " " T%! G.P

U = (−; +∞),

n=

+>K> !DM 9=7-`

D. .

=

lim

N →∞

lim

N →∞

N )

n

n=

−

−

N+

n+ =

=S + c "<−M Ld L p V8 1)

95 <#1 J

)

45 <#1 $ Q 8 J

∞ ) (− )n , n + sin x

U = [; π],

n=

&

∞ ) sin x sin(nx) √ , ) n+x

U = [; +∞],

n=

n=

sin(nx) , n

_+

n=

)

∞ ) n=

)

,

n

n

x + n(− ) , x +n

)

5& ! +P = =%! 88 G.P

R

&

∞ ) n= ∞ ) n=

=c ≥ max{a, b} T& P 88 G.P U &

n=

∞ ) (− )n+ ( + nx )n

n=

N ) sin(nx)

=

n=

≤

∞ )

=%! 88 G.P

[a; b] ⊆ R

n=

gn (x) =

I

I! &

fn (x) !

≤ | sin x |

√

fn (x)gn (x) =

∞ ) sin(nx) n

n=

&

∞ )

n sin

n= n

x

n

5& !

2 N

sin (x/ ) i: & L T%! 88 G.P x/n n ∞ {gn (x)}n= @QD V fn (x) = x (/) <

Ld &&x %! 88 G.P [a; b] & x &

∞ )

[a; b] fn (x)

n=

5& ! T&d Ld && T^&& =wc "<−M ∞ )

fn (x)gn (x) =

n=

∞ ) n=

n sin

x

n

="& 88 G.P U & T78 < &d Ld '. & PC. & " 5& ! 88 G.P +P " (

N S =

n=

& 5& !
U =

y !z & 5& ! L 4j(& 7.B L = 2+ & <: P&

Zj(& @ ! ^ L ! '. & + 2O5 "& ^ d +Q " =S <: @ ! ! L d & ^&& < 5 Z ] !

="& !

n→∞

∞ )

L P & {7; ^+.P =%! 88 G.P U & = %&E R − πZ @C. L [a; b] @C.

@QD .Px =%+ 88 G.P [; ] & ∞ w=+G& p

5& !

;

sin N + x × sin N x x sin

n=

x x x + + +· · · + x ( + x)( + x) ( + x)( + x)

an (x − x )n 8& n= < lim n |an |

π π

5& ! T78 Ld && mH

5& ! +P w 5& ! +P w 0

∞ )

U =

$ T%! 88 U &

x + nx )

84 9!0

"

@QD V gn (x) = < fn (x) = sin(nx) n k

+ i: w )

n

2

∞

n

&

5& !

{gn (x)}n=

|x| . n +x

nα (

n=

i: & L T%! 88 G.P

x , ( + x )n

< α ∞ )

∞ ) sin(nx) n

w="& xn = √ fn(x) N ^ +& .Px R

:Q )& C/I= !

fn (x)gn (x) 5& ! G d T"& 4 U& & 88

@C. &

U = [a; b] ⊆ R

n

∞ )

n=c

" ! 88 G.P @ T P + ( ∞ ) ) x+

5& ! S

P L & {gn (x)}∞ @ QD T"& 88 U n=b C ∞ G.P U & {gn (x)}n=b 5& @QD < "& 8 x ∈ U

U = R,

)

fn (x)

n=a

∞ ) cos nπ , ) n + x n=

∞ )

∞ )

a lim n+ n→∞ an

! G.P 5" d m8C T"& B =V+P R . & < +

)

∞ ) cos(nx) , n

n=

1

U = [a; b] ⊆ R,

45 <#1 $ Q 8 J

95 <#1 J

lim G.P ! G d T|x| < S mH = = n→∞ = T

R T|x| = S T =%! S< ! G d T|x| > S < %!

=S<

∞ )

(− )

n=

T
<

n

; )

∞ )

n

n=

! < P < x

= ±

G d

L %! 2DC ! G.P @ mH

G d T[a; b] ⊆ (− ; ) S 88

+ x + x + · · · + xn + · · · −−−−−−→

$ =+G& p

∞ ) n=

xn n!

−x

!

[a; b] &

2 N

/(n + )! = = lim R n→∞ /n! R

& ! ^ 5 =%!

^

∞ ) xn =f (x) = n!

∞

∞ ∞ ) ) nxn− xn = = f (x) f (x) = n! n! n=

x

+ ···+

<

∞ )

an xn

n=

S

n=

88 xn + · · · −−−−−−→ ex , [a; b] & n!

C/ ( 2 Q

= f (x)

U& & (−R; R) @L& &

+ i:

y = f (x) & "& R G.P 5" & ! ∞ )

bn xn

n=

=

x+

x

G.P y &

+ ···+

x n+ + ··· (n + )!

! ∞ )

n + (−)n (x +

n=

n

2 N

= af (x)

R

=

=

+ (−) )/(n + ) n n n→∞ ( + (−) )/n n ⎞ ⎛ − − n ⎝ lim n ⎠ = lim n→∞ n + n→∞ + − lim

(

n+

G.P ! G d T|x +

n+

S ^&& =R = mH S T S < %!

G d Tx + = S =x + = ± G d T|x + | = ∞ ) − n ( +( L T%! S< V ) ) C ! n n= | <

min{R, r} ≤

&

G.P 5" &

R

G.P 5" &

∞ )

∞ ) n=

∞ )

an xn

n=

& 24

= g(x)

aan xn

w

=%!

(an + bn )xn

n=

w0

=%! G.P y = f (x) + g(x) G.P 5" &

∞ )

(a bn + a bn− + · · · + an b )xn

n=

w>

=%! G.P y = f (x)g(x) & min{R, r} ≤

)n

$ =+G& p

n=

< %! G.P

^ =%! N+N C a < "& G.P y

ex − e−x

bn xn

C/ ( S 2:8 +

^+t.P sinh x =

∞ )

=an = bn n P L & 24 ^ =w < R

"& r G.P 5" & !

G d T"& (Q a < b S T5 =f (x) = ex + +x+

S

G d T"& R G.P 5" G.P x L & ! G d T|x − x | < R S w[Q =%! M7; =%! S< x L & ! G d T|x − x | > R S w U & ! G d TU = [a; b] ⊆ (x − R; x + R) S w| =%! 88 G.P

! G.P 5" mH = < f () = 24

an (x − x )n

n=

x "& G.P y

V+ i: =%! 88 G.P

n=

∞ )

& ! '

T" !

G d TU =

∞ ) n=

an xn

L

x |x| < R ,

∞ )

bn xn

n= ∞ )

x

an |x|n < r

n=

& S w9 < |a | < r

=& P G.P U & 4 ! !

∞ )

an xn

n=

+ i:

) /: 9$G

< %! G.P y = f (x) & "& R G.P 5" & ! |x − a| < R x P L & G d =−R < a < R 3 f (a) n ="& G.P y = f (x) & _+ ∞ n= n! (x − a) (n)

p

∞ ) n=

xn

!

2

!

$ =%! ' && R ! ^ G.P 5" =+G& 10

95 <#1 J

45 <#1 $ Q 8 J

: C & < R

=

=%! S<

lim n+ |a

n+ | n+ |a |

n→∞

= × × · · · × (n + ) =%! G.P R & ! mH =R = ∞ 4 P T^&& L %DC Q5 B . mH

=

n+

f (x) = ( − a )

∞ ) (− )n x n × n!

n=

n

+ a

∞ ) (− )n n! x (n + )!

n=

2+ N

x−

"& ∞ )

∞ ) n=

()

= =

-

x

= (n + )!

n=

n(n − )an xn− + x

∞ )

nan xn− +

n=

2# N

2 N

a

n

= =

a

n+

lim

= =

π √

∞ )

an xn =

+

+ a =

^&& +

V . mH

x→−

+

S

((n + )(n + )an+ + nan + an ) xn + a + a =

==

n=

=

=+G& p

n=

,G p ^ & =

dx + x (x + ) lim ln x − x + x→− √ arctan x√− +

⎧ ⎪ ⎪ a = − a , ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ a = − a = − ( − a ), ⎪ ⎪ ⎪ ⎪ a , a = − a = ⎪ ⎪ ⎪ ⎩=

n→∞ n+

C ! N

−

n

(n + )an+ + an =

∞ ∞ ) ) (− )n (− )n n+ = lim x n + n + x→− n= n= $ - x #) ∞ = lim (−x)n dx x→−

:= y + xy + y = + @Q5 ∞ ) an xn V+ i: d B ' f (x) =

n ≥ P L & < a

n |an | n→∞ n n+ (− ) = lim (n + )! n→∞

∞ ) (− )n n + n= - dx V+ D!3 x +

)n )/

ε

lim

=V+ D!3

=

n=

^ G.P 5" V+ B 7 ^ & !H & /& R & O& < + < %! ∞ && ! L T"&

lim

C ! G d Tx +

& mH =w r4 Q5 5x

x (− )n x n+ x + − ··· + + ··· sin x = x − ! ! (n + )!

=

−

n

2f.B & ! ! ' L T%! S< _+ ^ V |x+ | < ! G.P @ ^&& =w-$x %! Q<_ +l − ="& ; −

%" d V+!H

R

n=

( +(

'+ P ! <

n=

x (− )n x n+ x + − ···+ ! ! (n + )!

=

n

n ∞ ) − (− )n + n

n+

=%! 7+ s& T ! +Q "< L 8 f (x) = sin x U& n @D 7+ s& V+ ,e 5& & %! && x = @;N

∞ )

V B

√ ln π = +

=%! " ! J= =J L w0x %./ L w x 8 Y+

∞ )

an xn

n=

(− )n ( − a ) × × · · · × n (− )n ( − a ) n × n! (− )n a × × · · · × (n + ) (−)n n! a (n + )!

! G.P 5" @D!3 & |
R

= =

1>

lim

n→∞ n

|a

n

n|

| − a | √ = n ( n!)/

45 <#1 $ Q 8 J

)

∞ )

−n +

n

n=

95 <#1 J

e−nx ,

∞ ) n=

(− )n x n (n + )(n + )

n=

f (x) = ax ,

)

f (x) = cos x,

)

f (x) = sinh x,

)

f (x) = cosh x,

)

f (x) = arctan x,

)

f (x) = arcsin x,

)

f (x) = sin x, x

) f (x) = √

− x

,

x , + x − x

)

f (x) =

)

f (x) = ( + x)e−x ,

)

f (x) = e−x , ⎛

)

f (x) = ln⎝

)

f (x) =

)

f (x) = ex cos x.

⎞ + x⎠ , −x

x , ( − x)

x + ×

) )

+

x−

x

+

x

x

+ @Q5

&&x x = − L & < %! G.P ln & x = L &

vX < , ="& S< w^&& < %! '+ P ! V
x + ··· ×

∞ ) xn (n!)

n=

n=

xn (n)!

! +P w>

x − ln(x + ) − x

) )

cosh x = ln

+

= r4 y() = y

!

+ ···+

+

x

=

∞ ) n=

(− )n xn+ . (n + )(n + )

x

+

= x

∞ ) n=

(− )n x n . (n + )(n + )

' P G.P @ < G.P 5" "
) ) )

n

x + ··· (n)!

)

n=

x n+ x + ···+ + ··· ! (n + )! x

x

! +P w>0

•

sinh x = x +

(x − ) ln(x + ) −

Tx & mK! < V+h. V+N x & ^+:X %! : , x ∈ (− ; )∪(; ) P L & + =V+P& / x

P /& @ mK! < . %&E L < L ' P
& %D ;& ^ ^+:X L V+S

n=

= r4 xy + y − y =

+ @Q5

x

Ǳ_B @C/ '. &x [; x] @L& & d %!& < L x ∈ (− ; ] P L & + =V+S ,G wǱ_B &

− ···

∞ )

@L& & <

n=

+ ···

x + ×

]

x+

(− ; ) &

∞ ) (− )n xn+ 88 −−−−−−→ x ln(x + ), (− ; ] & n+

x + + ··· + ! ! x

,

∞ ) (− )n xn+ 88 −−−−−−→ x ln(x + ), (− ; ] & n+

)

x

x+

88

,G [;

+ D!3 L ! . )

− x + x − · · · + (− )n xn + · · · −−−−−−→

)

2% N

=
∞ ) m(m − ) · · · (m − n + ) n x , ) n!

5" mK! T : L U& L ' P Q ' s& + D!3 74 ! G.P

! ;N U&

x n+ x +x + ···+ + ··· = x + −x n +

) 19

∞ ) n= ∞ ) n= ∞ ) n= ∞ ) n=

n! n x , an

(a > ), n

an b + n n

xn ,

) )

∞ )

n +

n=

xn ,

n

∞ ) (n!) n x , (n)!

n=

∞ ) n=

n +

+x −x

n ,

∞ ) xn √ , ) a n

xn , an + b n

n=

( + (− ) ) n x , n n n

)

∞ ) (− )n n n n x , n! e

n=

95 <#1 J

45 <#1 $ Q 8 J

F=9=J L wIx %./ & +D" "< L mK! < %! w=+ <"

∞ ) xn ex = n! n=

n

! & =+h. !

24 ^ T3+34 C q < p + i: w>1 +P xp−

+x

+ xq

+

dx =

x +x

p

−

p+q

G d T− x +

+

p + q

−

p + q

5+DX C n < . Sn 4 w>I +P T"& S Sn S + + ···+ + ··· = ! ! n!

e

,5 " n; V8 +P .Px

+· · ·

< x < S +P x + · · · = + −x + x + x

w>J

∞ ) n(n + )(n + ) = n!

n=

16

e

45 <#1 $ Q 8 J

95 <#1 J

1F

'"* [12] Dieudonne, J., Linear Algebra and Geome-

[1] Adams, R. A., Calculus of Several Variables,

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11

A.,

Geometry,

Mir Pub.,

)+($ )J T

(αx + β)n ax + bx + c )J TP (x) ax + bx + c

6 TG.P +GQG 8" & U& L

9J TG.P−ε 9J T ! Ld

JF T5& ! & c "<−M 61 T c "<`M 6J T0 c "<`M JF T5& ! & &d 1F TC ! & &d 16 TC ! & ,G 60 T ! ,G %De U& IJ TC ! & .C @7.B 1> TC ! & DO JI T5& ! & 78 16 TC ! & 78 6I T ! ,G 78 1I TC ! & & 1> TC ! & " @ IJ TC ! &

IJ TC ! & "

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