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s = s(t) v(t) = s(t) a(t) = v(t) ! " # $
% % #& '
# ( a = a(t) %
a(t) # v(t) a
v # s )
%
% " #% * # +',( * " *
% -
% % #
% %# % % % % .
% % % /
% % 0 *
& #& * ! % % #
. Δ % %
f F % Δ
F f
Δ F Δ F (x) = f (x) x ∈ Δ
F (x) = x3
f (x) = x2 . #
%
# ! % %
2x sin x1 − cos x1 , x = 0, f (x) = x = 0. 0, & 2 x sin x1 , x = 0, F (x) = x = 0. 0, 3
f Δ f Δ
/ " % %%
! Δ F
G f " ! ! !# Δ # F G f # G(x) = F (x) + C,
x ∈ Δ,
C $ # 1 F
f F (x) =
f (x) Δ F (x)+C
f (F (x) + C) = F (x) = f (x) 1
F G # %
f F (x) = G (x) = f (F (x)−G(x)) = F (x)−G(x) = 0 2 3 F (x) − G(x) = C Δ
% # " " f Δ
f ! f (x) dx
. " # f
dx 4 %
& # - f % % % f (x)dx %% % 1 F
f "
'5(
f (x) dx = F (x) + C.
2 '5(
F $ dF (x) = F (x)dx = f (x)dx.
. f # F Δ dF (x) = F (x) + C F (x)dx = F (x) + C 4
% & # .
f1 f2
# % Δ (f1 + f2) # Δ
(f1(x) + f2 (x)) dx =
f1(x) dx +
f2(x) dx.
'6(
. # .
F1 F2 # %
f1 f2 F1 (x) = f1(x), F2 (x) = f2(x) f1 (x) dx = F1(x) + C1,
f2(x) dx = F2(x) + C2,
C1, C2
% % . F (x) := F1(x) + F2(x) - F
(f1 + f2) F (x) = F1 (x) + F2 (x) = f1 (x) + f2 (x),
x ∈ Δ.
2
(f1(x) + f2(x)) dx = F (x) + C = F1(x) + F2 (x) + C,
C
2 %
f1(x) dx +
f2(x) dx = F1 (x) + C1 + F2 (x) + C2 .
. C, C1, C2
% % F1(x) +F2(x) +C F1(x) +C1 +F2(x) +C2 # '6( . λ λ = 0
λf (x) dx = λ
'7(
f (x) dx,
# . F (x)
f (x) λF (x)
λf (x) (λF (x)) = λF (x) = λf (x) 8
λf (x) dx = λF (x) + C1 ,
C1
2 %
λ
f (x) dx = λ(F (x) + C2 ) = λF (x) + λC2 ,
C2
. λ = 0 C1 C2
% % λF (x) + C1 λF (x) + λC2 # '7( ' # . λ1, λ2 %
(λ1f1 (x) + λ2 f2 (x)) dx = λ1
f1(x) dx + λ2
f2(x) dx.
9
6 7
5 xα dx = xα+1 + C, α = −1. 6 dxx = ln |x| + C. 7 ax dx = lna a + C, a > 0, a = 1, ex dx = ex + C : sin x dx = − cos x + C. ; cos x dx = sin x + C < cosdx x = tg x + C. = sindx x = − ctg x + C. > x dx+a = a1 arctg xa + C = − a1 arcctg xa + C, a = 0 a = 0. ? x dx−a = 2a1 ln x−a x+a + C, 5@ √adx−x = arcsin xa + C = − arccos xa + C, |x| < |a|. √ 55 √xdx−a = ln |x + x2 − a2| + C, |x| > |a|. √ 56 √xdx+a = ln |x + x2 + a2| + C. 1 %
& % % % " % % & α+1
x
2
2
2
2
2
2
2
2
2
2
2
2
5 (3 sin x + 5 cos x) dx ( ) / " # ' :(
(3 sin x+5 cos x) dx = 3
6
sin x dx+5
√ √ ( x + x) x dx
cos x dx = −3 cos x+5 sin x+C.
( ) A % " #
√ √ ( x + x) x dx =
x dx +
5
x2 2x 2 + + C. x dx = 2 5 3 2
√ dx. 2. 3 x dx. 3. √dxx . 4. (x3 + 3x) dx. 5. 2 sin2 x2 dx. √ (2+x)3 1−x 2 √ 2 √ dx. dx. 7. ( x + 3)(x + x) dx. 8. 6. x x 5·2xx+2·5x 2 9. dx. 10. tg x dx. 2x
1.
.
f (x) ϕ(t) % Δx
Δt ϕ(Δt) = Δx % f (ϕ(t)), t ∈ Δt . ϕ & ϕ−1(x) Δx
* # ϕ Δt
f (x) Δx ϕ(Δt) = Δx f F Δx
f (x) dx =
f (ϕ(t))ϕ(t) dt
t=ϕ−1 (x)
.
# -
F
f
F = f . x = ϕ(t) F (ϕ(t))
f (ϕ(t))ϕ(t) 9 [F (ϕ(t))] = F (ϕ(t))ϕ(t) = f (ϕ(t))ϕ(t).
f (x) dx = F (x) + C = F (ϕ(t))|t=ϕ−1 (x) + C =
C
f (ϕ(t))ϕ(t) dt
t=ϕ−1 (x)
,
8 % #
f (ϕ(t))ϕ(t) dt =
f (ϕ(t)) dϕ(t) =
f (x) dx
.
x=ϕ(t)
4 %
5 e5x dx ( ) 2 t = 5x /% x x = 15 t, 1 5 dt .
1 e dx = 5 5x
dx =
1 1 et dt = et + C = e5x + C. 5 5
6 tg x dx ( ) B
sin x dx = − cos x
tg x dx =
d cos x = − ln | cos x| + C. cos x
7 x x+1 dx ( ) 8 xdx = 12 dx2 = 12 d(x2 + 1) 2
x 1 dx = x2 + 1 2
d(x2 + 1) 1 = ln(x2 + 1) + C. 2 x +1 2
/ % t = x2 + 1
√ 3 11. cos 3x dx. 12. lnxx dx. 13. x9 7 8 − x10 dx. 14. x2x+4 dx. e2x 3 x √dx dx. 17. . 18. 15. a−2x dx. 16. arctg 2 2 ex +1 dx. sin x 1−ctg x cos x 1+x 19. √3 2 dx. 20. √1+x dx. 1−x2 sin x
!
u(x) v(x)
Δ v du u dv ! +
u dv = uv −
v du.
# .
d(uv) = v du + u dv
*
u dv = d(uv) − v du.
B & % & 5 d(uv) = uv + C & # 8 & u dv " # ' 6( u dv =
d(uv) −
v du = uv −
v du
v du,
C . & B #& " / P (x) cos ax dx, P (x) sin ax dx, P (x)eax dx, P (x) a = 0
u P (x)
sin ax, cos ax, eax . " B ' :( * P (x) xn n A %
%" % ' % # ($
1 x cos ax dx = a n
1 n x d sin ax = xn sin ax − a a n
xn−1 sin axdx.
B
n x
5 x sin 3x dx ( ) . x sin 3x dx = −
1 3
1 1 xd cos 3x = − x cos 3x + 3 3
cos 3xdx =
1 1 = − x cos 3x + sin 3x + C. 3 9
6 x2ex dx ( ) / u = x2 dv = ex dx v = ex,
2 x
x e dx =
2
2 x
x de = x e − 2 x
du = 2xdx
xex dx.
. u = x
dv = ex dx xex dx = x dex = xex − ex dx = xex − ex + C.
!
x2ex dx = x2ex − 2xex + 2ex + C.
"" /
P (x) arcsin ax dx,
P (x) arccos ax dx, P (x) arctg ax dx, P (x) arcctg ax dx P (x) ln x dx % P (x)dx dv B
% 1
%
% m, m > 0
m
m C %& P (x) xn n
1 x ln x dx = n+1 n
m
ln x dx m
n+1
xn+1 m m ln x− = n+1 n+1
xn lnm−1 x dx.
5 x ln x dx ( ) / u = ln x dv = xdx v =
x2 , 2
du =
1 x
dx
2 x2 x 1 x2 1 x2 x2 x ln x dx = ln x − · dx = ln x − x dx = ln x − + C. 2 2 x 2 2 2 4 6 arccos 2x dx 1 ( ) . u = arccos 2x dv = dx v = x, du = −2 √1−4x
2
. $
dx
x √ dx = 1 − 4x2 1 1 2 −1/2 2 = x arccos 2x − (1 − 4x ) d(1 − 4x ) = x arccos 2x − 1 − 4x2 +C. 4 2 arccos 2x dx = x arccos 2x + 2
""" B% ebx sin ax dx, ebx cos ax dx,
% . # ebx $
I1 :=
1 e sin ax dx = b bx
sin ax de
bx
a = 0, b = 0
1 a = ebx sin ax − b b
ebx cos ax dx.
. # ebx 1 a I1 = ebx sin ax − 2 b b
cos ax debx =
1 bx a a2 bx = e sin ax − 2 cos ax e − 2 ebx sin ax dx. b b b . I1 I1 1 bx a a2 bx I1 = e sin ax − 2 cos ax e − 2 I1 . b b b
/% # I1
ebx e sin ax dx = 2 [b sin ax − a cos ax] + C. a + b2 bx
)
ebx cos ax dx =
ebx [b cos ax + a sin ax] + C. a2 + b2
2 −x 21. x cos 2x dx. 22. arcsin x dx. 23. x e dx. 24. 3 x 2 ln x dx. √ x 25. x3 dx. 26. arctg x dx. 27. x e dx. 28. ln x dx. 2 29. x tg x dx. 30. cos ln x dx.
9 % % " * %
A # % # %$ an xn + an−1xn−1 + ... + a0 R(x) = , bN xN + bN −1xN −1 + ... + b0
an = 0, bN = 0 1 " n < N % 1 " n ≥ N % ' % & % ( . Pn(x) QN (x) R(x) n ≥ N R(x) =
Pn (x) Tk (x) = S(x) + , QN (x) QN (x)
S(x) Tk (x) % k < N 9 # % " % . " %# A , (x − a)k
Bx + C , (x2 + px + q)m
p2 − 4q < 0, k, m ∈ N,
&% .
# " %
A, B, C
(x) $ # # #
# * # PQ(x)
, r
Q(x) =
(x − aj )
kj
j=1
s
(x2 + pl x + ql )ml ,
l=1
aj $ ! ! Q(x) kj , j = 1, ..., r- "! x2 + plx + ql ! p2l − 4ql < (k ) (1) 0, l = 1, ..., s . ! Aj , ..., Aj , j = (1) (m ) (1) (m ) 1, ..., r; Bl , ..., Bl , Cl , ..., Cl , l = 1, ..., s ! j
l
P (x) = Q(x) +
s
l=1
r
j=1
l
(1)
(k )
(2)
Aj Aj Aj j + + ... + x − aj (x − aj )2 (x − aj )kj
(1) (1) Bl x + C l x 2 + p l x + ql
+
(2) (2) Bl x + C l (x2 + pl x + ql )2
+ ... +
+
(m ) (m ) Bl l x + C l l (x2 + pl x + ql )ml
.
4x −3x A " (x−2) (x +1) 2
2
2
( ) 9 * 4x2 − 3x A1 A2 Bx + C = + . + (x − 2)2(x2 + 1) x − 2 (x − 2)2 x2 + 1
. # & # 4x2 − 3x A1(x − 2)(x2 + 1) + A2(x2 + 1) + (Bx + C)(x − 2)2 = . (x − 2)2(x2 + 1) (x − 2)2(x2 + 1)
D % % % A1, A2, B, C
4x2 − 3x = A1 (x − 2)(x2 + 1) + A2 (x2 + 1) + (Bx + C)(x − 2)2. ':(
9 % * % % . * * %
: :
% $ ⎧ ⎪ ⎪ ⎨
A1 + B −2A1 + A2 − 4B + C A1 + 4B − 4C ⎪ ⎪ ⎩ −2A1 + A2 + 4C
= 0 = 4 = −3 = 0
!# A1 = 1, A2 = 2, B = −1, C = 0 ' % ':( x = 2 % A2( 4x2 − 3x 1 2 x = + − (x − 2)2(x2 + 1) x − 2 (x − 2)2 x2 + 1
" B % # # # " A B " (x−a) % . n = 1 n
. n > 1
A dx = A ln |x − a| + C. x−a
A dx = A (x − a)n
(x − a)−n d(x − a) = −
A + C. (n − 1)(x − a)n−1
2 B " (x Bx+D +px+q) p − 4q < 0
(t bt+c +a ) % 2
2
n
2 n
p 2 p2 x + px + q = x + +q− , 2 4 2
% t = x + p2 ' a2 q − p4 ( . n = 1 2
bt + c dt = b t2 + a2
t dt+c 2 t + a2
t2
t 1 b c dt = ln(t2 +a2 )+ arctg +C. 2 +a 2 a a
. n > 1 B (t bt+c +a ) dt .% $ 2
2 n
t b dt = − + C. 2 n +a ) 2(n − 1)(t2 + a2 )n−1 9 In := (t2+a1 2)n dt % #
& % In In−1$ 2 1 t + a2 − t2 1 In := dt = 2 dt = (t2 + a2 )n a (t2 + a2 )n 1 1 1 t = 2 dt − dt = t a (t2 + a2 )n−1 a2 (t2 + a2 )n 1 t 1 1 1 = 2 In−1 − 2 − + dt = a a 2(n − 1)(t2 + a2 )n−1 2(n − 1) (t2 + a2 )n−1 1 t 1 = 2 In−1 + 2 In−1. − 2 2 2 n−1 a 2a (n − 1)(t + a ) 2a (n − 1) b
(t2
t 1 In = 2 + 2a (n − 1)(t2 + a2 )n−1 a2
1 1− In−1, 2(n − 1)
n = 2, 3, ... .
I1 % % I2,
I3
';(
5 x 3x+5 +2x+5 dx ( ) 9 " * % % x2 +2x+5 = (x+1)2 +4 2 #
t = x + 1 ' * dx = dt( 2
3x + 5 dx = x2 + 2x + 5
3t + 2 dt = 3 t2 + 4
t dt + t2 + 4
2 dt. t2 + 4
/ t % . $ 3 3 t x+1 ln(t2 + 4) + arctg + C = ln(x2 + 2x + 5) + arctg + C. 2 2 2 2
6
3x+5 x2 +2x−3
dx
( ) % / 9 "
# x = 1, "
x = −3
A
A B 3x + 5 = + . (x − 1)(x + 3) x − 1 x + 3
. & & # 3x + 5 = A(x + 3) + B(x − 1). 4 % x * x = −3 B = 1 x = 1 A = 2 3x + 5 dx dx dx = 2 + = 2 ln |x − 1| + ln |x + 3| + C. x2 + 2x − 3 x−1 x+3
% 0 /% % x2 + 2x − 3
(x + 1) − 4 2
2 # t
3x + 5 dx = x2 + 2x − 3
3t + 2 dt = 3 t2 − 4 t
= x+1
' *
t dt + t2 − 4
= dx = dt(
2 dt. t2 − 4
/ % . $ 3 3 1 t − 2 1 x − 1 2 + C = + C. ln |t2 − 4| + ln ln |x ln + 2x − 3| + 2 2 t + 2 2 2 x + 3
. % "
ln |x2 + 2x − 3| = ln |(x − 1)(x + 3)| = ln |x − 1| + ln |x + 3|; x − 1 = ln |x − 1| − ln |x + 3|. ln x + 3
. *
3x + 5 dx = 2 ln |x − 1| + ln |x + 3| + C. x2 + 2x − 3
31. 35. 39.
5 4 x+1 x −8 x dx. 32. x2 −3x+2 dx. 33. xx+x dx. 3 −4x dx. 34. (x−1)(x+3) x4 −3x2 +2 2 3 2 2 2x −7x dx x −2x +3x−6 x (x−2)2 (x+1) dx. 36. x(x2 +1) . 37. x3 −3x2 +4 dx. 38. 1−x4 dx. x3 +x+1 dx dx. 40. (x2 +2)2 (x2 +1)(x2 +x) .
!
"#!$
. R(u1, ..., un) u1 = f1(x), ..., un = fn (x) R(f1(x), ..., fn(x)) f1(x), ..., fn(x) " A %
ax + b R x, cx + q
r1
ax + b , ..., cx + q
rn
dx.
'<(
0 r1, ..., rn % % ri = mp m pi % aq −bc = 0 ' ( 2 '<( tm = ax+b cx+q /% qt −b =: ρ(t) - ρ # x x = a−ct #E
ρ # . * r ax+b dx = ρ(t) dt cx+q = (tm) = tp '<(
$ i
m
m
i
ax + b R x, cx + q
r1
ax + b , ..., cx + q
pi m
rn
i
dx =
R(ρ(t), tp1 , ..., tpn )ρ(t) dt.
! % c = 0 % '<(
R x,
√
r1
ax + b, ...,
√
rn
ax + b dx.
5 1+dx√x ( ) 2 x = t2
dx = 2t dt .
2t dt t + 1 − 1 dt dx √ = =2 = 1+t 1+t 1+ x √ √ dt = 2 dt − 2 = 2t − 2 ln |1 + t| + C = 2 x − 2 ln |1 + x| + C. 1+t 1−x 6 1+x dx
2 ( ) 2 1−x 1+x = t /% # x
x=
1−t2 1+t2
−4t dx = (1+t 2 )2 dt 2 1−x −4t t +1−1 dx = t dt = −4 dt = 1+x (1 + t2 )2 (1 + t2 )2 1 1 = −4 dt + 4 dt. 1 + t2 (1 + t2 )2
.% % ';( a = 1, n = 2 .
1 t + −4 arctg t + 4 2(1 + t2) 2
1 2t dt = −2 arctg t + + C. 1 + t2 1 + t2
. x $
1−x dx = −2 arctg 1+x
1−x + 2 (1 + x)(1 − x) + C. 1+x
41. 45. 48.
x+ √3 2+x dx x+1 dx √ √ √ √ √ dx. 42. . 43. dx. 44. . 3 x( x−3) x− 3 x x x−2 2+x √ √ 6 x+3−1 dx √ √ dx. 46. x√x−1 dx. 47. 3 x−2 3 x−1 (x−1)3 . x+3(1+ x+3) x+1 √ √x dx√ √ √x+2 √ dx. 49. . 50. dx. 3 2 √ x(1+ 3 x)3 ( x+2+ x−1)(x−1)2 x −4x
"" 1 % x2 + px + q &%
# r %
R(x,
r
x2 + px + q) dx
'=(
%& # 9
R(x, r x2 + px + q) = R(x, r (x − a)(x − b)) = 1/r x−a x−a = R x, |x − b| r = R1 x, . x−b x−b
/ % % '=( A
√
% R(x, ax2 + bx + c) dx a = 0 % % % . ! C% x2 + px + q % % % # t2 ± a2 9 %
R(t, t2 ± a2 ) dt,
R(t,
a2 − t2 ) dt
'>(
#& % √ ( R(t, a2 − t2) dt, −a ≤ t ≤ a t = a · sin y − π2 ≤ y ≤ π2 E √ ( R(t, t2 − a2) dt, |t| ≥ a t = sina y − π2 ≤ y ≤ π2 , y = 0E √ ( R(t, t2 + a2) dt t = a · tg y − π2 < y < π2
√
1 − x2 dx, −1 ≤ x ≤ 1 ( ) 2 x = sin y, −π/2 ≤ y ≤ π/2
1 = 2
dx = cos y dy
2 1 − x2 dx = 1 − sin y cos y dy = cos2 y dy =
1 1 y 1 [1 + cos 2y] dy = dy + cos 2y dy = + sin 2y + C = 2 2 2 4 y 1 1 1 = + sin y cos y + C = arcsin x + x 1 − x2 + C. 2 2 2 2
1 ! "!
A % 0, a = 0
√ Ax+B ax2 +bx+c
dx A =
9 * %
# &
& %
A (2ax + b) + B − Ab Ax + B 2a 2a √ √ dx = dx = 2 2 ax + bx + c ax + bx + c A d(ax2 + bx + c) dx Ab √ √ = dx + B − . 2a 2a ax2 + bx + c ax2 + bx + c
.%
% % % %
! % a > 0
Ax + B √ dx = ax2 + bx + c b 1 2 2aB − Ab A 2 √ ln |x + +√ ax + bx + c + ax + bx + c| + C. = a 2a 2a a a ! % a < 0 ' b2 − 4ac > 0 % " # x( Ax + B √ dx = ax2 + bx + c −2ax − b A 2 2aB − Ab √ arcsin √ = ax + bx + c + + C. a 2a −a b2 − 4ac
√−x3x+4 dx +6x+8 ( ) /%
# % −2x + 6 2
− 32 (−2x + 6) + 13 3x + 4 √ √ dx = dx = −x2 + 6x + 8 −x2 + 6x + 8 d(−x2 + 6x + 8) dx 3 √ dx = dx + 13 =− 2 −x2 + 6x + 8 17 − (x − 3)2 x−3 = −3 −x2 + 6x + 8 + 13 arcsin √ + C. 17
P 2 " , A (x)dx
Pn(x) n B & √ n ax2 +bx+c
Pn (x)dx √ = Qn−1 ax2 + bx + c + λ 2 ax + bx + c
√
dx , ax2 + bx + c
Qn−1 (n − 1) % * λ 9
& #
* % Qn−1 λ
√xx −x−1 dx +2x+2 ( ) . 3
2
x3 − x − 1 √ dx = (ax2 + bx + c) x2 + 2x + 2 + λ x2 + 2x + 2
9
√
1 dx. x2 + 2x + 2
x3 − x − 1 √ = x2 + 2x + 2
x+1 λ = (2ax + b) x2 + 2x + 2 + (ax2 + bx + c) √ +√ . x2 + 2x + 2 x2 + 2x + 2
! $
x3 − x − 1 = (2ax + b)(x2 + 2x + 2) + (ax2 + bx + c)(x + 1) + λ.
2 * % % x ⎧ ⎪ ⎪ ⎨
3a 5a + b 4a + 3b + c ⎪ ⎪ ⎩ 2b + c + λ
= 1 = 0 = −1 = 1.
! a = 13 ,
b = − 56 , c = 16 , λ = 52 ! x3 − x − 1 √ dx = x2 + 2x + 2 1 5 1 2 dx = = (2x − 5x + 1) x2 + 2x + 2 + 6 2 (x + 1)2 + 1 1 5 = (2x2 − 5x + 1) x2 + 2x + 2 + ln(x + 1 + x2 + 2x + 2) + C. 6 2
% # B (x−α) √dxax +bx+c k
2
7 x − α = 1t 9 dx = − dtt ax2 + bx + c = (aα +bα+c)tt +(2aα+b)t+a x > α, t > 0 2
2
dx √ =− (x − α)k ax2 + bx + c
2
2
tk−1
dt. (aα2 + bα + c)t2 + (2aα + b)t + a
dx (x−1)√−x +2x+3 ( ) . x − 1 = 1t x = 1t + 1 2
dx √ =− (x − 1) −x2 + 2x + 3
1 t
dx = − dt t2
B
dt t2
=
1 t)
+ 2(1 + + 3 dt dt 1 1 1 2 =− = − ln t + t − + C = =− √ 2 2 4 1 4t2 − 1 2 t −4 √ 2 1 2 + −x + 2x + 3 = − ln + C. 2 2(x − 1)
√ 51.
55. 59.
1+x2 x4
dx. 52.
2 √2x −3x x2 −2x+5 √ 6x+5 x2 −4x+8
√
dx. 56. dx. 60.
dx . (x2 −4)3
53.
√
dx √ . (x+2)2 x2 +5 √x dx. x− x2 −1
−(1 +
1 2 t)
3 − 2x + x2 dx. 54.
57.
2 √x +5x+3 5−4x−x2
dx. 58.
√ 8x−11 5+2x−x2
dx.
√dx . x2 x2 −2x
""" A %
(a + bxβ )α xγ dx,
a, b &% α, β, γ % . % % % % 2 x = t ' dx = β1 t −1dt( 1 β
1 β
(a + bxβ )α xγ dx =
1 β
(a + bt)α tλ dt,
λ = γ+1 − 1 β A 5 α . λ = mn m n > 0 % 2 F u = t
1 n
6 λ . α = mn m n > 0 % 2 F u = (a + bt)
7 α + λ . %" α = mn m n > 0 % B 1 n
(a + bt)α tλ dt =
a + bt t
α
tα+λ dt.
2 F 8 u = a+bt 1/n t 8 % * %
√ 3 √ 1+ 4 x
√x dx. A" 2 t = x x = t4, 4t3dt 1 4
dx =
√ 3 1 1 1 1 1+ 4x √ dx = x− 2 (1 + x 4 ) 3 dx = 4 t(1 + t) 3 dt. x
8 α = 13 , λ = 1 * 6 2
u = (1 + t) t = u3 − 1, dt = 3u2du 1 3
√ 3 1+ 4x 12 √ dx = 12 (u6 − u3 ) du = u7 − 3u4 + C = 7 x
=
%
7 4 1 7 1 4 12 12 (1 + t) 3 − 3(1 + t) 3 + C = (1 + x 4 ) 3 − 3(1 + x 4 ) 3 + C. 7 7
&! "#!
" B R(sin x, cos x)dx
u = tg x2 , −π < x < π .
2du * x = 2 arctg u, dx = 1+u
% # #& $ 2
sin x =
2u , 1 + u2
cos x =
1 − u2 . 1 + u2
9 cos2 x2 x x x sin x =
B
2 sin 2 cos 2 2 tg 2 2u = = ; 1 + tg2 x2 1 + u2 cos2 x2 + sin2 x2
cos2 x2 − sin2 x2 1 − tg2 x2 1 − u2 cos x = = = . 1 + tg2 x2 1 + u2 cos2 x2 + sin2 x2
R(sin x, cos x)dx =
1 − u2 2u , R 1 + u2 1 + u2
2du . 1 + u2
8 %
R(sin x, cos x) dx %# % u = sin x, u = cos x, u = tg x /
&# * " %
&# 1 R(u, v)
u R(−u, v) = R(u, v) % R(u, v) = R1(u2, v) & " % u 1
R(u, v)
R(−u, v) = −R(u, v) % R(u, v) = u · R1(u2, v)
%& R(u,v) u ( R(− sin x, cos x) = −R(sin x, cos x) % R(sin x, cos x)dx = R1(sin2 x, cos x) sin x dx = −R1(1−cos2 x, cos x)d cos x,
u = cos xE ( R(sin x, − cos x) = −R(sin x, cos x) % u = sin xE ( R(− sin x, − cos x) = R(sin x, cos x) % u uv v u u u R(u, v) = R( v, v) = R1 ( , v) = R2 ( , v 2 ), v v v
R1( uv , −v) = R1( uv , v) R1 ( . * R(sin x, cos x) = R2(tg x, cos2 x) = R2
1 tg x, , 1 + tg2 x
u = tg x '−π < x < π(
dx 5 1−sin x ( ) . % # u = tg x2
6
dx 2du 2du = = = 2u 1 − sin x 1 − 2u + u2 (1 + u2)(1 − 1+u 2) 2 = −2 (1 − u)−2d(1 − u) = 2(1 − u)−1 + C = + C. 1 − tg x2 sin x (1−cos x)2 dx
( ) . %
* u = cos x
sin x dx = (1 − cos x)2
d(1 − cos x) = (1 − cos x)2
du = u2
1 1 = − +C =− +C u 1 − cos x
"" B% sinm x cosn x dx,
n, m % % # m
n
( n m % u = cos 2x % % . m = 2k + 1, n = 2 + 1
1 sin2k x cos2 x sin 2x dx = x cos x dx = sin 2 k 1 + cos 2x 1 − cos 2x 1 d cos 2x = =− 4 2 2 2k+1
2+1
=−
1 2k++2
(1 − u)k (1 + u) du.
( n m % u = tg x % % . * x = du E arctg u, dx = 1+u 2
u2 1 1 1 − u2 = sin x = (1 − cos 2x) = ; 1− 2 2 1 + u2 1 + u2 2 1 1 1 − u 1 = 1+ . cos2 x = (1 + cos 2x) = 2 2 1 + u2 1 + u2 0 n m %
sin2 x = 1−cos2 2x , cos2 x = 1+cos2 2x
2
* % " ( n m u = cos x n m u = sin x
5 tg4 x dx ( ) 2 u = tg x /% x x = arctg u, du dx = 1+u 2
tg4 x dx =
u4 du. 1 + u2
/% 5 $ 4 u −1+1 du 2 du = (u − 1) du + = 1 + u2 1 + u2 u3 tg3 x = − u + arctg u + C = − tg x + x + C. 3 3
6 sin2 x dx ( ) /% % $
sin2 x dx =
1 1 − cos 2x dx = 2 2
1 dx − 2
cos 2x dx =
x sin 2x − +C. 2 4
sin x 7 cos dx x ( ) D * ' u = cos x( * % 3
2
sin2 x 1 − cos2 x d cos x = − d cos x = cos2 x cos2 x 1 − u2 1 du 1 =− + u + C = + cos x + C. du = − + du = u2 u2 u cos x
sin3 x dx = − cos2 x
""" / sin ax cos bx dx, sin ax sin bx dx,
$ cos ax cos bx dx
1 sin ax cos bx = [sin(a + b)x + sin(a − b)x]; 2 1 sin ax sin bx = [cos(a − b)x − cos(a + b)x]; 2 1 cos ax cos bx = [cos(a + b)x + cos(a − b)x]. 2
sin x cos 2x dx ( ) / "
% %" 1 [sin 3x − sin x] dx = sin x cos 2x dx = 2 1 1 1 1 sin 3x dx − sin x dx = − cos 3x + cos x + C. = 2 2 6 2
61.
65. 68.
1+ctg x dx dx dx 4 . 3+2 cos x . 62. 5−4 sin x+3 cos x . 63. 1−ctg x dx. 64. 1−sin x 3 sin x 2 3 2 5 cos8 x dx. 66. sin x cos x dx. 67. sin x cos x dx. dx . 69. sin 2x sin 3x dx. 70. cos x cos2 3x dx. sin4 x cos2 x
/ # *
% * %
*
. % G & G e dx = x t dt % E ln sin x dx % E x −x e dx, % E dx √ √ x dx √ dx
'0 < k < 1 (1−x )(1−k x ) (1−x )(1−k x ) (1+hx ) (1−x )(1−k x ) h
% ( * % t
2
2
2
2 2
2
2 2
2
2
2 2
'
√ 3
√ 4 x 1.x + C. x4 + C. 3.2 x + C. 4. x4 + ln3 3 + C. 5.x − sin x + C. √ √ √ 2 6. − x1 − 2 ln x + x + C. 7. 27 x7 + x3 + x2 + 2 x3 + C. 8. − √16x + 24 x + √ √ x 4 x3 + 25 x5 + C. 9.5x + ln25 52 + C. 10. tg x − x + C ' $ 2. 34
% cos2 x( 2
7 7 11. 13 sin 3x + C. 12. 12 ln2 x + C. 13. − 80 (8 − x10)8 + C. 14. 12 x2 − √ −2x 1 4 C. 16. arctg x + C. 17.2 1 − ctg x + C. 2 ln(x2 + 4) + C. 15. − 2a ln a + 4 √ √ 3 x x 18.e − ln(e + 1) + C. 19.3 sin x + C. 20.√arcsin x − 1 − x2 + C. 21. 12 x sin 2x+ 14 cos 2x+C. 22.x arcsin x+ 1 − x2 +C. 23.−(x2 +2x+ √ 2)e−x + C. 24.x ln x − x + C. 25. ln13 x3x − ln12 3 3x + C. 26.(x + 1) arctg x − √ x + C. 27.(x3 − 3x2 + 6x − 6)ex + C. 28.x ln2 x − 2x ln x + 2x + C. 2 29.x tg x − x2 + ln | cos x| + C. 30. x2 (cos ln x + sin ln x) + C ' $
"
(
31. 12 ln |x − 1| + 12 ln |x + 3| + C. 32. − ln |x − 1| + 2 ln |x − 2| + C. 3 2 33. x3 + x2 + 4x + 2 ln |x| + 5 ln |x − 2| − 3 ln |x + 2| + C. 34. 12 ln |x2 − 2| − 1 2 ln |x2 −1|+C. 35. ln |x−2|+ln |x+1|+ x−2 +C. 36. ln |x|− 12 ln(x2 +1)+C. 2 37.x+ 37 ln |x−2|− 34 ln |x+1|+C. 38. 14 ln |x+1|− 41 ln |x−1|− 21 arctg x+C. 1 1 2 √ 39. 4(x2−x arctg √x2 +C. 40. ln |x|− 12 ln |x+1|− 14 ln(x2 + 2 +2) + 2 ln(x +2)− 4 2 1) − 12 arctg x + C √ √ √ √ 1 3 41. 2 x − 2 + 2 arctg √x−2 + C. 42. − ln |x| + ln | x − 3| + C. 3 2 √ √ √ √ 6 3 6 43. 23 (x + 2)3 − 4 √ x + 2 + 65 √ (x + 2)5 + C. 44. 2 x + 3 x + 6 x+ √ √ √ 3 6 3 6 6 6 ln | x−1|+C. 45. 3 x + 3−6 x + 3−3 ln | x + 3+1|+6 arctg x + 3+ √ √ √ √ C. 46. ln( x + 1 + x − 1) − ln( x + 1 − x − 1) − 2 arctg x−1 x+1 + C. 7 3 3 x−2 4 1 x+2 2 x+2 2 x+2 47. − 37 3 x−2 x−1 + 4 x−1 +C. 48. − 3 · x−1 + 3 x−1 − 3 ln 1 + x−1 + √ √ √ √ √ 12 23√ x+3 6 6 5 12 12 5 12 3 +ln |x|−3 ln |1+ x|+C. 50. x + x + ln | x5 − C. 49. 32 · (1+ 3 x)2 5 5 5 1| + C 1 1 51. − 3 sin3 (arctg + C. 52. − 4 sin(arccos + C. 53. 2 arcsin x+1 2 2 + ) x) x √ x−1 2 √ + C. sin(2 arcsin x+1 2 ) + C. 54. − 8 5 + 2x − x − 3 arcsin 6 √ √ √ x2 +5 2 − 27 ln |5− 55. x x2 − 2x + 5−5 ln |x−1+ x2 − 2x + 5|+C. 56. − 9(x+2) √ √ 2 2x+3 x2 + 5|+ 27 ln |x+2|+C. 57. − x2 + 2 5 − 4x − x2 + 32 arcsin x+2 3 + 3 1 √ C 58. − 16 1 − x2 2 + 12 1 − x2 2 + C 59. 6 x2 − 4x + 8 + 17 ln |x − 2 + √ 3 x2 − 4x + 8| + C. 60. x3 + 13 (x2 − 1)3 + C ' $
x +
√
x2 − 1( 1 61. √25 arctg 63. ln | sin x − cos x| + C. + C. 62. 2−tg x + C. 5 2 √ 64. 12 tg x+ 2√1 2 arctg( 2 tg x)+C. 65. 7 cos1 7 x − 5 cos1 5 x +C. 66. x8 − sin324x +C. 1 1 1 1 cos 2x − 64 cos2 2x + 96 cos3 2x + 128 cos4 2x +C. 68. − 3 tg13 x − tg2x + 67. − 32 1 1 1 sin 5x + C. 70. 12 sin x + 20 sin 5x + 28 sin 7x + C. tg x + C 69. 12 sin x − 10 tg x2 √
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