高数积分公式大全
常 用 积 分 公 式
(一)含有ax b +的积分(0a ≠) 1.
d x ax b +?=1
ln ax b C a ++
2.()d ax b x μ
+?
=
11
()(1)
ax b C a μμ++++(1μ≠-)
3.
d x x ax b +?=21
(ln )ax b b ax b C a +-++
4.2d x x ax b +?
=22311()2()ln 2ax b b ax b b ax b C a ??
+-++++????
5.
d ()x
x ax b +?=1ln ax b C b x +-+
6.
2d ()
x
x ax b +?
=
21ln a ax b C bx b x +-++ 7.
2
d ()x x ax b +?=21(ln )b ax b C a ax b
++++ 8.22d ()x x ax b +?
=2
31(2ln )b ax b b ax b C a ax b
+-+-++ 9.
2
d ()
x
x ax b +?
=211ln ()ax b C b ax b b x +-++
的积分
10.
x C +
11.x ?=2
2
(3215ax b C a -
12.x x ?=2223
2(15128105a x abx b C a
-+
13.
x
?
=22
(23ax b C a -
14
.
2x ?
=222
3
2(34815a x abx b C a -+ 15
.?
(0)
(0)
C b C b ?+><
16
.
?
2a bx b -- 17.
d x x ?
=b ?18.
2d x x ?
=2a + (三)含有2
2
x a ±的积分 19.
22d x x a +?=1arctan x C a a
+ 20.
22d ()n x x a +?=2221222123d 2(1)()2(1)()n n x n x
n a x a n a x a ---+-+-+? 21.
22d x
x a -?=1ln 2x a C a x a -++
(四)含有2
(0)ax b a +>的积分
22.2d x ax b +?
=(0)
(0)
C b C b ?+>+<
23.
2d x x ax b +?=2
1ln 2ax b C a
++
24.22d x x ax b +?=2d x b x
a a ax
b -+?
25.2
d ()x x ax b +?=2
21ln 2x C b ax b
++ 26.
22d ()x x ax b +?=21d a x
bx b ax b --+?
27.32d ()x x ax b +?=2222
1
ln 22ax b a C b x bx +-+
28.
22d ()x ax b +?=221d 2()2x x
b ax b b ax b +++?
(五)含有2
ax bx c ++(0)a >的积分
29.2d x ax bx c ++?
=22(4)
(4)
C b ac C
b a
c +<+>
30.
2d x x ax bx c ++?=2
21d ln 22b x ax bx c a a ax bx c
++-++?
(0)a >的积分 31
.
?
=1arsh
x
C a
+
=ln(x C ++ 32
.
C +
33
.
x ?
C
34.
x =C +
35.
2
x 2ln(2a x C +
36.
2
x =ln(x C +++
37.
?1C a +
38.
?2C a x -+
39.
x 2ln(2
a x C ++
40.x =2243(25ln(88
x x a a x C ++
41.x ?C +
42.x
x ?
=422(2ln(88
x a x a x C +++
43.
x ?a C +
44.
x ?
=ln(x C +++
(0)a >的积分
45.
?
=
1arch x x
C x a
+=ln x C ++
46.
C +
47.
x ?
C
48.
x =C +
49.
2
x 2ln 2a x C +++
50.
2
x =ln x C +++
51.
?1arccos a
C a x
+
52.
?C +
53.
x 2ln 2
a x C -++
54.x =2243(25ln 88
x x a a x C -++
55.x ?C
56.x
x ?=422(2ln 88
x a x a x C -+
57.
x ?
arccos a a C x -+
58.
2
d x x ?
=ln x C x
-+++
(0)a >的积分 59.
?
=arcsin
x
C a
+ 60.
C +
61.
x ?
=C +
62.
x C +
63.
2
x =2arcsin 2a x C a + 64.
2
x arcsin
x
C a
-+
65.
?1C a +
66.
?C +
67.
x 2arcsin 2a x C a
+
68.x =2243(52arcsin 88x x a x a C a
-+
69.x ?=C
70.x
x ?
=422(2arcsin 88x a x x a C a
-+
71.
x ?a C +
72.
x ?
=arcsin x
C a
-+
(0)a >的积分
73.
?
2ax b C +++
74.
x
2
2ax b C +
+++
75.
x ?
2ax b C -
+++
76.
?
=C +
77.
x 2
C +
78.
x ?
=C ++
79.
x ?
=((x b b a C --+
80.
x ?=((x b b a C -+-
81.
?
=C ()a b <
82.
x 2()arcsin 4b a C -+ ()a b < (十一)含有三角函数的积分 83.sin d x x ?
=cos x C -+ 84.cos d x x ?
=sin x C + 85.tan d x x ?=ln cos x C -+ 86.cot d x x ?=ln sin x C + 87.sec d x x ?=ln tan(
)42
x
C π++=ln sec tan x x C ++ 88.csc d x x ?
=ln tan
2
x
C +=ln csc cot x x C -+ 89.2
sec d x x ?=tan x C + 90.2
csc d x x ?
=cot x C -+ 91.sec tan d x x x ?=sec x C + 92.csc cot d x x x ?
=csc x C -+
93.2
sin d x x ?=
1
sin 224x x C -+ 94.2
cos d x x ?=1sin 224
x x C ++
95.sin d n
x x ?=1211sin cos sin d n n n x x x x n n ----
+? 96.cos d n x x ?=12
11cos sin cos d n n n x x x x n n
---+? 97.d sin n x x ?=121cos 2d 1sin 1sin n n x n x
n x n x ----?+
--? 98.d cos n x x ?=121sin 2d 1cos 1cos n n x n x
n x n x
---?+--? 99.cos sin d m n x x x ?=11211cos sin cos sin d m n m n
m x x x x x m n m n
-+--+++? =112
11cos sin cos sin d m n m n n x x x x x m n m n +----+++?
100.sin cos d ax bx x ?
=11
cos()cos()2()2()a b x a b x C a b a b -
+--++-
101.sin sin d ax bx x ?
=11
sin()sin()2()2()
a b x a b x C a b a b -
++-++-
102.cos cos d ax bx x ?
=
11
sin()sin()2()2()
a b x a b x C a b a b ++-++-
103.
d sin x
a b x +?
tan
x
a b C ++22()a b >
104.d sin x a b x +?
C
+22()a b <
105.
d cos x
a b x +?
)2
x C +22()a b >
106.d cos x a b x +?
C +22()a b <
107.
2222d cos sin x a x b x +?=1arctan(tan )b
x C ab a + 108.
2222d cos sin x
a x
b x -?=1tan ln 2tan b x a C ab b x a ++-
109.sin d x ax x ?=
211
sin cos ax x ax C a a -+ 110.2
sin d x ax x ?=223122cos sin cos x ax x ax ax C a a a -+++
111.cos d x ax x ?=211
cos sin ax x ax C a a ++
112.2
cos d x ax x ?=223122sin cos sin x ax x ax ax C a a a
+-+
(十二)含有反三角函数的积分(其中0a >)
113.arcsin d x x a ?=arcsin x x C a
++
114.arcsin d x
x x a ?=22()arcsin 24x a x C a -+
115.2
arcsin d x x x a
?=3221arcsin (239x x x a C a ++
116.arccos d x
x a ?
=arccos
x
x C a
117.arccos d x
x x a ?=22()arccos 24x a x C a -
118.2
arccos d x x x a
?=3221arccos (239x x x a C a -+
119.arctan
d x x a ?=22
arctan ln()2x a x a x C a -++ 120.arctan d x x x a ?=22
1()arctan 22
x a a x x C a +-+
121.2
arctan d x
x x a
?=33222arctan ln()366x x a a x a x C a -+
++ (十三)含有指数函数的积分