常用求导积分公式及不定积分基本方法

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常用求导积分公式及不定积分基本方法

This model paper was revised by LINDA on December 15, 2012.

一、基本求导公式

1. ()1x x μμμ-'= ()ln 1x x

'= 2. (sin )cos x x '= (cos )sin x x '=-

3. 2(tan )sec x x '= 2(cot )csc x x '=-

4. (sec )tan sec x x x '= (csc )cot csc x x x '=-

5. ()ln x x a a a '=,()x x e e '=

6. ()

2arctan 11x x '+= ()arcsin x '= ()

2arccot 11x x '+=- ()arccos x '= 二、基本积分公式 1. 1d (111)x x x C μμμμ+=+ =-/ +⎰, 1ln ||+dx x C x

=⎰ 2. d ln x

x

a a x C a =+⎰,d x x e x e C =+⎰ 3. sin d cos x x x C =-+⎰, cos d sin x x x C =+⎰

4. 2sec d tan x x x C =+⎰ 2csc d cot x x x C =-+⎰

5. tan d ln |cos |x x x C =-+⎰ cot d ln |sin |x x x C =+⎰

6. sec d ln |sec tan |x x x x C =++⎰ csc d ln |csc cot |x x x x C =-+⎰

7.

2

1d arctan 1x x C x =++⎰ arcsin x x C =+

2211d arctan x x C a x a a =++⎰ arcsin x x C a =+

8.

ln x x C =+

(

ln x x C =++ 9. 221

1d ln 2x a

x C a x a x a -=+-+⎰

三、常用三角函数关系

1. 倍角公式

21cos 2sin 2x

x -= 21cos 2cos 2x

x +=

2. 正余切与正余割

正割 1

sec cos x x = 22sec 1tan x x =+

余割 1csc sin x x = 2

2csc 1cot x x =+

四、常用凑微分类型 1.

1

1

()d d ()ln ()()()f x x f x f x C f x f x '==+⎰⎰;

2.1()d ()d() (0)f ax b x f ax b ax b a a

+=++≠⎰⎰; 3.11()d ()d (0)f x x x f x x μμμμμμ-⋅=≠⎰⎰; 4.1()d ()d (0,1)ln x x x x f a a x f a a a a a

=>≠⎰⎰; (e )e d (e )de x x x x f x f =⎰⎰; 5. 1(ln )d (ln )d ln f x x f x x x

⋅=⎰⎰; 6. (sin )cos d (sin )dsin f x x x f x x = ⎰⎰;

(cos )sin d (cos )dcos f x x x f x x =-⎰⎰;

7. 2(tan )sec d (tan )d tan f x x x f x x =⎰⎰;

2(cot )csc d (cot )dcot f x x x f x x =-⎰⎰;

8.(sec )sec tan d (sec )dsec f x x x x f x x ⋅=⎰⎰;

(csc )csc cot d (csc )dcsc f x x x x f x x ⋅=- ⎰⎰; 9.

(arcsin )(arcsin )d arcsin f x x f x x = ⎰⎰;

21(arctan )d (arctan )d arctan 1+f x x f x x x

⋅= ⎰⎰. 五、第二类换元法常用的代换方法

t a x sin =; (2) 22x a +,可作代换t a x tan =;

(3) 22a x -,可作代换t a x sec =;

(4) 分母中次数比较高时,常用倒代换代换1x t

=;

,可作代换t =

t = 六、分部积分

基本公式 udv uv vdu =-⎰⎰

基本方法: ()f x dx ⎰

()()()f x u x v x '=−−−−−→分解()()u x v x dx '⎰−−−→凑微分()()u x dv x ⎰ −−−−→分部积分()()()()u x v x v x du x =-⎰

使用分部积分法的关键是将()f x dx 恰当地凑成()()u x dv x 的形式,其遵循的一般原则是:

(1)()v x 容易求得;(2)()()v x du x ⎰要容易积分;

一般地,按“反 对 幂 指 三”的顺序,前者取为)(x u ,后者取为()v x '. 反三角函数 对数函数 幂函数 指数函数 三角函数

1. ()11cos 2d cos 22d cos d()2222x x x x x x x '=⋅=⎰⎰⎰ (1cos d 2u u ⎰

) 1sin 22

x C =+ 2. ()331(25)d (25)25d 2

x x x x x '+=+⋅+⎰⎰ 31(25)d(25)2x x =++⎰ (31d 2

u u ⎰) 41(25)8

x C =++ 3. ()22222

2d d d x x x xe x e x x x e '=⋅=⎰⎰⎰ (d u u e u e C =+⎰) 2

x e C =+

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