常用的不定积分公式

kdx kx c

=+?

1

1x x dx c

μμ

μ+=++?

ln dx

x c

x =+? ln x

x

a a dx c

a =+? x x

e dx e

c

=+? 21

arctan 1dx x c x =++?

arcsin x c

=+ 2211arctan x

dx c a x a a

=++?

2211ln 2x a dx c

x a a x a -=+-+?

arcsin x c

a =+

ln x c

=

cos sin xdx x c

=+? sin cos xdx x c

=-+? 2

21sec tan cos dx xdx x c x ==+?? 2

21csc cot sin xdx x c

x ==-+?? tan ln cos xdx x c =-+? cot ln sin xdx x c =+? sec ln sec tan xdx x x c =++? csc ln csc cot xdx x x c

=-+?

积分型

换元公式

()()()1

f ax b dx f ax b d ax b a +=

++?

? u ax b =+ ()

()()

11

f x x dx f x d x μμμμμ

-=?? u x μ=

()()()1

ln ln ln f x dx f x d x x ?=??

ln u x = ()()()x x x x f e e dx f e d e ?=?

?

x u e =

()()()1

ln x x x x f a a dx f a d a a

?=

?

? x u a =

()()()sin cos sin sin f x xdx f x d x ?=?? sin u x =

()()()cos sin cos cos f x xdx f x d x ?=-?? cos u x =

()()()2tan sec tan tan f x xdx f x d x ?=?

?

tan u x = ()()()2

cot csc

cot cot f x xdx f x d x ?=??

cot u x = ()()()2

1

arctan arc n arc n 1f x dx f ta x d ta x x ?

=+?

? arctan u x =

()()()

arcsin arcsin arcsin f x f x d x =?

?

arcsin u x =

⑴形如n ax

x e dx ?

，令n u x =，ax

dv e dx =

形如sin n

x xdx ?令n u x =，sin dv xdx =

形如cos n

x xdx ?

令n u x =，cos dv xdx =

⑵形如arctan n

x xdx ?

，令arctan u x =，n dv x dx =

形如ln n

x xdx ?

，令ln u x =，n dv x dx =

⑶形如sin ax e xdx ?，cos ax

e xdx ?

令,sin ,cos ax

u e x x =均可。