考研数学必备积分表【附录】

5 1 .
dx
1 ln
|x|
C
x x2 a2 a a x2 a2
x2 a2 | C
5 2 .
dx
1 arccos a C
x x2 a2 a
x
A．6 含有a bx cx2 的积分
57.
sin
axdx
1 a
cos
ax
C
53.
dx a bx cx2
2 arctan 2cx b C(b2 4ac) 58.
a
2 6 .
a2 x2
3
2dx
x
5a 2 2 x2
8
a 2 x 2 3a 4 arcsin x C
8
a
19. dx 1 ln a x C
a2 x2 2a a x
2 7 .
dx a2 x2
3 2
a2
x
C
a2 x2
2 8 .
xdx 3
a 2 x 2 2
dx
4c
8c
a bx cx2
56.
xdx 1 a bx cx2 b
dx
a bx cx2 c
2c a bx cx2
61.
sin 2
axdx
x 2
1 4a
sin
2ax
C
62.
cos2
axdx
x 2
1 4a
sin
2ax
C
63.
sec
axdx
1 a
ln
|
sec
ax
tan
ax
|
C
64.
x
a
ln
a2 x2
C
a
a2
87.
xn
arcsinxdx
1
n
1
xn1
arcsin
x
x n 1
dx
1 x2
88.
xn
arctanxdx
1
n
1
x n 1
arctan
x
x n 1 1 x2
dx
A．10 几个常用的定积分
89. cos nxdx sin nxdx 0
90. cos mx sin nxdx 0
a2 x2 2a 2x 2
1 2a3
ln
a
a2 x2 C x
3 3 .
a2 x2 dx
a 2 x 2 a ln a
a2 x2 C
x
x
3 4 .
a2 x2
1
x2
dx x
a 2 x 2 arcsin x C a
A．5 含有 x2 a2 的积分
42.
dx
3
x2 a2 2
8
8
45.
38. dx ln | x x2 a2 | C
x2 a2
39. xdx x2 a2 C
46.
x2 a2
40. x2dx x x2 a2 a2 ln | x x2 a2 | C
47.
x2 a2 2
2
41.
x2 a2
3
2 dx
x
2x2 5a2
a bx C(a 0) a
6.
dx
x a bx
1 ln a
x a bx
C
14. x2
dx a bx
a bx ax
a 2b
x
dx a bx
7.
dx
x2 a bx
1 ax
b a2
ln
a bx x
C
15.
a bx dx 2 x
a bx a x
dx a bx
8.
dx
x a bx2
a
a
79.
x
ln
xdx
x 1
12
1 ln
x
1
C(
1)
80. xn sin xdx xn cos x n xn1 cos xdx
81. xn cos xdx xn sin x n xn1 sin xdx
82.
eax
sin
bxdx
eax
a sin bx b
a2 b2
cos bx
2a3
|x|
x3
dx
x2 a2
x2 a2 2a2x2
1 2a3
arccos a x
C
x 2 a 2 dx x 2 a 2 a ln a x 2 a 2 C
x
|x|
x 2 a 2 dx x 2 a 2 a arccos a C
x
x
5 0 .
x2 x2
a2
dx
x2 a 2 ln | x x
1 b3
1 2
a
bx 2
2a
a
bx
a2
ln
|
a
bx
|
C
9. x a bxdx
15b2
C
3
2 15b2x2 12abx 8a2 a bx2
10. x2 a bxdx
105b2
C
11.
xdx
2bx 2a a bx
C
a bx
3b2
12.
x2dx
2 3b2 x2 4abx ቤተ መጻሕፍቲ ባይዱ8a2
8
a
18
xdx (a2 x2 )n
1 2
ln
a2 x2 1
2(n 1) a2
C(n 1) x2 n1 C(n 1)
2 4 .
xdx a 2 x2 C
a2 x2
2 5 .
x 2dx x a 2 x 2 a 2 arcsin x C
a2 x2
2
2
1
C
a2 x2
A．4 含有 a2 x2 的积分
2 9 .
x 2d x
3
a 2 x 2 2
1
arcsin x C
a2 x2
a
3 0 .
dx
1 ln a a 2 x 2 C
x a2 x2 a
x
3 1 .
dx
x2 a2 x2
3 2 .
dx
x3 a2 x2
a2 x2 C a2x
C
83. eax cos bxdx eax a cos bx b sin bx C
a2 b2
84. arcsin xdx x arccos x a2 x2 C
a
a
85. arccos xdx x arccos x a2 x2 C
a
a
86.
arctan
x dx
x
arctan
sin n
xdx
n
1 n
n n
3 2
4 5
2 3
(n是奇数）
n
1 n
n n
3 2
3 4
1 2
2
(n是偶数）
95.
2 0
sin 2 m1
x cosn
xdx
2462m
n 1n 3n
2m 1
96.
2 0
sin 2 m
x cos2n
xdx
1352n 11352m 2462m 2n
csc
axdx
1 a
ln
|
csc
ax
cot
ax
|
C
65. sec x tan xdx sec x C
A．8 含有三角函数的积分
66. csc x cot xdx csc x C
67.
sin
ax
sin
bxdx
sin a b x 2a b cosa b x
sin a b 2a b cos a b
1
a a bx
1 a2
ln
a bx x
C
16.
a x2
bx
dx
a
x
bx
b 2
x
dx a bx
A．3 含有a2 x2 的积分
2 0 . a 2 x 2 d x x a 2 x 2 a 2 a r c s in x C
2
2
a
17.
dx a2 x2
n
1 arc tan x C(n 1)
0(m n)
91. cos mx cos nxdx (m n)
0(m n)
92. sin mx sin nxdx (m n)
0(m n)
93. sin mx sin nxdx
cos
mx
cos
nxdx
0
0
2 (m n)
94. 2 cosn xdx 0
2 0
简单积分表
A．2 含有 a bx 的积分
A．1 含有a+bx 的积分
3
23bx 2aa bx2
1.
a
bx dx
(a bx) 1 b( 1)
C (
1)
1
b
ln
|
a
bx
|
C(
1)
xdx x a 2. a bx b b2 ln | a bx | C
3.
x2dx a bx
x2 a2 3a4 ln | x
48. x2 a2 | C
8
8
49
x2dx 3
x2 a2 2
x ln | x x2 a2
x2 a2 | C
dx
x2 x2 a2
x2 a2 a2x2
C
dx
x 2 a 2 1 ln a x 2 a 2 C
x3 x2 a2
2a2x2
1
2
4ac b2
4ac b2
1
ln
b2
4ac b 2cx
C(b2
59.
4ac)
b2 4ac b2 4ac b 2cx
60.
cos axdx 1 sin ax C a
tan axdx 1 ln | cos ax | C a
cot axdx 1 ln | sin ax | C
mn
mn
77.
a
dx b cos
x
2
a2 b2 arctan
a a
b b
tan
x 2
C(a2
b2 )
1
bacos x ln
b2 a2 sin x C(a2 b2)
b2 a2
a bcos x
A.9 其他形式的积分
78. xneaxdx 1 xneax n xn e 1 axdx
1
tann1 x
tann2 xdx(n 1)
n 1
69.
sin a b x sin a b x cos ax cos bxdx 2a b 2 a b C(a b)
73. cotn xdx 1 cotn1 x cotn2 xdx(n 1) n 1
74. secn xdx 1 tan xsecn2 x n2 secn2 xdx(n 1)
a bx
15b3
a bx C
xdx
4. a bx2
1 b2
a a bx
ln | a bx | C
5.
x2dx
a bx2
x b2
b3
a2
a
bx
2a b3
ln
|
a
bx
|
C
13. x
dx
1
ln |
a bx
a | C(a 0)
a
a bx a
a bx
2 arctan a
a
a
x
2(n 1)a2 a2 x2 n1
2n 3 2(n 1)a2
2 1 . x
a2 x2 dx 1
3
a2 x2 2 C
3
dx
a2 x2
22. x2 a2 x2 dx x 2x2 a2
n1
(n
1）
23.
dx
8 arcsin x C
a2 x2
a
a 2 x 2 a 4 arcsin x C
a
A．7 含有 a bx cx2 的积分
54.
dx a bx cx2
1 c
ln | 2cx
1
arcsin
b2 2cx
c b
a bx cx2 | C(c 0) C(b2 4ac,c 0)
c
b2 4ac
55. a bx cx2dx 2cx b a bx cx2 4ac b2
a2
x
C
x2 a2
35. x2 a2 dx x x2 a2 a2 ln | x x2 a2 | C
2
2
43.
xdx 3 x2 a2 2
1 C x2 a2
36.
x
x2 a2 dx 1
3
x2 a2 2 C
44.
3
37. x2 x2 a2 dx x 2x2 a2 x2 a2 a4 ln | x x2 a2 | C
x
x
C
(a
b)
70.sinn 71. cosn
xdx 1 sinn1 xcos x n1 sinn1 xdx
n
n
xdx 1 cosn1 sin x n1 cosn1 xdx
n
n
68. sin ax cos bxdx
2a b
2a b
C(a b) 72. tann xdx
n 1
n 1
75. cscn xdx 1 cot xcscn2 x n 2 cscn2 xdx(n 1)
n 1
n 1
76. sinm xcosn xdx sinm1 xcosn1 x n 1 sinm xcosn2 xdx
mn
mn
sinm1 xcosn1 x m1 sinm2 xcosn xdx