高数积分公式大全

（一）含有ax b的积分（a
1 . dx 1 ax b
a
=-In ax b
2.
3.
4.
5.
6.
7.
9.
10.
11.
常用积分公式
0)
1
(ax b) dx =
a( 1)
x 1
dx = -^(ax b
ax b a
丄dx =丄
ax b
dx
x( ax
b)
dx
x2(ax b)
x
2dx
(ax b)
a3
1ln
b
1
bx
ax
1
=-r(ln
a
(ax
bln
b)2
ax
b)
ax b) C
2b(ax b) b2ln
ax b
b
ax b)
ax b
2bln ax b
b2
ax b) C
dx
2
x(ax b) b(ax b)
含有ax b的积分
1
2 In
b2
ax b
Tax~ dx = —T(ax~b)3
3a
x、、ax bdx = -^(3ax 2b
15a
x2 . ax bdx = ^^(15a2x2 12abx 8b2) ., (ax b)3 C
105a
).(ax b)3 C
x 2
dx = - 2 (ax 2b)、ax b C ，ax b 3a
15. 16.
17. 18. 19. 20. 21 . (四) 22. 23.
14. dx x ¥ ax b
dx x 2
1 ax b ax b. dx = x ax b
2-'
x
含有x
2
dx
~2 x
(3a 2x 2 4abx 8b 2)、、ax 15a
2
- arctan ax b .b
C (b
C (b
0) 0)
ax b bx
2 ax b
ax b
2
a 的积分
_ dx (x 2
dx -2 x i 含有 a 2b
dx
2 . n 2 , 2 a ) 2(n 1)a (x 2
ax dx
2- ax
x
2
ax -dx b
|x a lx al C 1 ln 2a b(a 0)的积分 dx x, ax b
x 、ax dx b
2、n 1
a )
2n 3
2
2(n 1)a dx
T~2
2 n 1
(x a )
arcta n j a
x C
V b ax b
(b 0) 1
In 24 ab
丄
ln |ax 2a
C (b 0)
24. x 2—-
ax b dx = x a
dx ax 2
b
25. dx x(ax 2 b)
In
2b 2
x 2~
ax
26. dx x 2(ax 2 b) 1 bx
27.
a l 3 2 2 ln
x 3(ax 2 b)
2b 2
dx
3
28. dx
2 2
(ax b)
29.
30. (六) 31.
32. 33. dx ~2- ax
2
ax b 2
x 2b(ax 2 b)
1
2b
dx ax 2 b
含有ax
2
bx dx ax 2
bx c
x
dx ax bx c 含有■.
x 2
dx dx
.(x 2 a 2)
X
7x 2" c(a 0)的积分
-=2 2 arctan —2ax b _ C ■. 4ac b 、4ac b
a 2 (a 丄 ln ax
2a arsh- a
—dx = 2 a
dx =
34. 2 ax b J b 2
4ac
2 ax b J b 2
4ac
C b
.b 2
4ac (b 2 4ac)
2 (b 4ac) 2 bx c 0)的积分 dx ___
2a ax 2
bx c
C 1 = ln(x x 2 a 2) C
a 2 C
35. 36. 2
—x ______ d x =
7^ d
,(x a ) ln( x \ x 2 a 2)
37. dx x i x 2 a 2 lln a
dx —
2 2
.x a
2 a x
39. x 2
a 2
dx =
40. (x 2
a 2
)3
dx
41 . X' x 2 a 2
dx =
38. 42.
2 .“ 2 x x a
a 2)
45. 46. 47.
x
-(2x 8 3
= x 2
2 a
x x 2 a 2
2 x
dx = 43. dx = 44. 含有\ x 2
■- x 2 a 2
5a 2
)、. x 2
2
a
2 2
a
ln( x . x 2 a 2
) C
4
a In(
x
8
、.x 2 a 2) C
,V x 2
a a In 2 2
x
—— In (x
a 2
(a 0)的积分
、x
2
a 2)
dx x 2 a 2
dx .(x 2
2\3
a )
x
arch
凶 a ____ x a 2 x 2
C 1 =ln
x
x :=a
2 dx =
含有、•. a
2
x 2
(a 0)的积分
dx
x
2
2、3
2 2
2
x ) a ■ a x
48.
x 」
1
dx =
C
J\3 I 2
2
a )
x a 49. 2
x
2 2
x a 2 r a a ln
2 50. 2
x . dx = 2 2\3 x a ) x . ln 2 2 x a
x 2
a
51 .
dx
1
=—arccos
x 、x * 2 a 2 a
dx x 2
、x 2
a 2
2 2
x a
2
a x 53. ；x 2
a 2
dx =
54. (x 2 a 2)3
dx
55. X x 2 a 2dx =
52. 56. x
—、x a
2
2
— In 2
x
2
2
a
57. 58.
=8(2x2 5a2)Q 2 ~2
, x 2 x x a dx =
(2x 8
~~2
2
x a 」 dx = x
a 2
)3
C
a
a arccos
— x
In x
4
—ln 8
I n
59.
=arcs in 仝
C a
60.
62.
65. 戸—dx= Ja2x2
.a x
______ dx
.(a2x2)3
C
63.
64.
2 2
a . x
arcs in
2 a
arcsin x C
斗—=ll n」a^ C
x、a2 x2 a x
dx
67. 、a2x2dx =
68.
(a2x2)3dx
69. x、a2x2dx =
70. x2 a2;x2dx
71 . a2
x
x2
dx
—
72. a2
x2
x2
dx
—
（九）含有■.ax2
66.
x2、aLx2
x.L
2 dx
2
a . x arcs in 2 a
ax2bx c =8(5a2 2x2)^^
x2)3 C
-2 2
.a x
a2 x2
bx c (a
I
n
2、 2 2
a ) . a x
3a4arcs in 仝C
8
4
a . x
arcs in
8 a
a 4a __x^
a In ----------
x
arcsin 一C
a
0）的积
分
2
ax
2- a
■ax2
bx c
= 2 ax b —2=
cdx = ------ ：，ax bx c
4a
.xdx = Sc bx ax 2
—空 arcsin —
2ax
—b — C
.c bx ax 2 a
2 . a 3
b 2 4ac
J(x
爲 x)
= 2arcSi
^H
C (a b)
2
3 4
2x a b / 、八 、(b a) .
..(x a)(b x)dx = --------- 乂x a)(b x)
------------------------- a rcsin
4
75. x ax 2
bx
dx =
76. dx c bx ax 2
77.
、c bx ax
2dx
4ac b 2
8 . a 3
In 2ax b 2石 bxc C
— In 2ax b 2^J a 4a 2府 1
2ax b
—i= arcsi n = C ■ a . b 4ac
2ax b ,
2
c bx ax 4a
x 2
bx c
b 2
4ac . 2ax b arcs inr 8 . a 3 b 2
4ac
74.
、.a?__bx
78.
80.
"F
(b
F C
82.
81.
(x a)(b x)的积分
79.
b 后（b 曲
C
4
(a b) （十一）含有三角函数的积分
sin xdx = cosx C cosxdx = sinx C
secx tan xdx = secx C
csc x cot xdx = cscx
83. 84. 85. 86.
87. 88.
89.
90. 91.
92.
93. 94. 95. 96. 97. 98. 99. =ln tan
— 2
C = ln cscx cotx C
=tanx C
cotx C
cscxdx sec xdx csc 2
xdx sin 2
xdx =
cos 2 xdx =
sin n
xdx =
cos n
xdx
=
dx
.n
sin x dx
x 1 sin 2x
2 4
x 1 . sin 2x 2 4
1 n 1 sin
xcosx n 1 n 1 cos xsi nx n
1 cosx n 1
1 sin x sin x
n
A
cos x n 1 m . n . cos xsin xdx
n 1
cos x
1
n 2
sin xdx cos n 2
xdx dx
.n 2-
sin x
dx
n~2~ n 1 cos x
m 1 . n 1 cos xsi n x m n
1 m 1 n 1 cos xsin m n
m cos 2
. n .
xsin xdx
m . n 2
-
cos xsin xdx
tan xdx = In cosx
cot xdx =
l n s i nx secxdx = ln
tan (—
|) C =In secx tanx
sin ax cosbxdx
=
sin ax sin
bxdx =
cosax cosbxdx =
dx
a bsin x
dx a bsin x dx a bcosx
cos(a b)x
2(a b) 1
sin(a b)x 2( a b)
1
si n(a b)x 2(a b) _1 ■- b 2 cos(a b)x C
2(a b) 1
sin (a b)x C
2(a b)
1
si n(a b)x 2(a b) b 2
b arctan --- 2 _ C (a 2
l
n 2 a
a 2
b 2
b 2
)
x ata n- 2
2 b
arctan(
a b \ a b
dx a bcosx
a
b 、b 2 a 2
丁脚；）
tan
仝
2
(a
(a 2 b 2
)
(a 2
b 2
)
2
b 2) dx
-2 2 ~2~~2~ a cos x b sin x
dx
~~2
2
7~2~~2~
a cos x
b sin x
1 xsin axdx = —^sin ax a x 2
sin axdx = 1 arctan(— tan x) ab a ab
丄ln
2ab
bta nx a
bta nx a
1 -x cos ax C
a 1 2 —x cosax a 2 — x sin ax a 2 —cosax a xcosaxdx = 2 cosax a
2 ,
1 2 ■
x cosaxdx = x sin ax a 1 . xsin
ax a 2
2 x cos
ax a
2 3
sin ax C
a
（十二）含有反三角函数的积分（其
中
100. 101. 102.
103.
104.
105.
106. 107.
10 8. 109. 110. 111. 112.
3
2 4 X x , x a 2
x arctan dx = arctan x a 3 a 6
含有指数函数的积分
a X dx = -^ a x
C
In a
ax
1
ax
e dx = e C
a ax 1 ax
xe dx = 2 (ax 1)e C a
n ax . 1 n n 1 ax ・
x e dx =—x e —X e dx
a a
X X 1 X
xa dx a 2 a C
In a (In a)
n x .
1
n X
n
n 1 x .
x a dx
x a
x a dx
In a In a
x arcs in -dx = xarcs in
a 2
• X, /X xarcs in dx =(— 2 x 2 arcsin x dx a arccos-dx = a 2
)arcs in 4 3 x . =—arcs in 3
x xarccos- a 2
X. /X x arccos —dx =(— a 2
-2
x
1
, 2 9(x 2
a x )arccos 一 4 a 4心2
x 2 C
2a) a 2
x 2 arccos —dx a 3
x =—arccos — a
arctan — dx = x arcta n — a xarctan x dx = -
(a 2
a 2 9(x 2 2a 2)、, a x 2
a 2
2 严x) C
x 2 )arctan
- a 113.
114.
115.
116.
117.
118.
119.
120. 121.
(十三) 122. 123. 124. 125.
126.
127. 12 8.
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3
a i / 2 ln(
x 2) C
e ax s in bxdx = —2 e ax(as inbx bcosbx) C
a b
129. 130. e ax cosbxdx = — 2 e ax
(bs inbx a b a
cosbx) 131. ax n e sin bxdx = a 2 1 b 2n 2 ax . n e sin 2
n(n 1)b
~2 .2 2
a bn bx(a s inbx
ax . n
e sin
2
bxdx
ax n
e cos bxdx =
——1 2 2 e ax cos n 1
bx(a cosbx a bn
C nbcosbx)
nbsin bx)
n(n 1)b 2 ax n 2…
2 -^2 e cos bxdx a bn
132. In xdx = xlnx x C dx .
133. -In In C
xln x n - 1 n 1
1 134. x In xdx
x (I x J C n 1 n 1
n 1 135. (In x)n
dx - x(In x)n n (In x) dx 136. x (In x) dx - 1 m
x 1/1 、
n n (In x) m 1 m 1
（十五） 含有双曲函数的积分 （十四）含有对数函数的积分 m
x (ln
137. shxdx = chx C x)n 1dx
13 8. chxdx = shx C 139. thxdx = In chx C 2
x 1
140. sh xdx = sh2x C 2 4 ,9 x 1 141. ch xdx = sh2x C 2 4 (十六)定积分 142. cosnxdx = sin nxdx = 0
143. cos mxs inn xdx = 0
,m n
0, m n
sin mxsin nxdx =
,m n
0, sin mxsin nxdx = cosmxcos nxdx =
0 0 _
2
= —C
144. cos mxcos nxdx
=
0, m n 145.
146.
2 = 2
sin
n
xdx 二 2 co n
xdx
n
n 1
n —
1
2
n
n 1 n 3 III 4 2
n
n n 2 II 5 3
n
为大于1 n 1 n
n 3 II I 3 1 （n 为正偶
数） n n 2 III 4 2 2
m n ,m n
的正奇数），h = 1
，I 0 = 2
147.。