微积分公式与定积分计算练习

微积分公式与定积分计算练习(附加三角函数公式)

一、基本导数公式

⑴()0c '= ⑵1x x μμμ-= ⑶()sin cos x x '=

⑷()cos sin x x '=- ⑸()2tan sec x x '= ⑹()2cot csc x x '=- ⑺()sec sec tan x x x '=⋅

⑻()csc csc cot x x x '=-⋅

⑼

()x

x

e e '= ⑽

()ln x

x a a a

'= ⑾

()1

ln x x '=

⑿

()

1log ln x a

x a '=

⒀

(

)arcsin x '= ⒁(

)arccos x '= ⒂

()21arctan 1x x '=+ ⒃()

21arccot 1x x '=-+⒄()1

x '=

⒅

'=

二、导数的四则运算法则

()u v u v '''±=± ()uv u v uv '''=+ 2u u v uv v v '''

-⎛⎫= ⎪⎝⎭

三、高阶导数的运算法则

(１)()()()

()

()

()

()

n n n u x v x u x v x ±=±⎡⎤⎣

⎦ (2)()()

()()

n n cu x cu x =⎡⎤⎣

⎦

(3)()()

()

()n n n

u ax b a u

ax b +=+⎡⎤⎣

⎦

(4)

()()()

()()()()0

n

n n k k k n k u x v x c u x v x -=⋅=⎡⎤⎣⎦

∑

四、基本初等函数的n 阶导数公式

(1)

()()

!

n n

x n =

(2)

()()

n ax b

n ax b

e a e ++=⋅

(3)

()()

ln n x

x n a a a

=

(4)

()()

sin sin 2n n ax b a ax b n π⎛⎫+=++⋅⎡⎤ ⎪⎣⎦

⎝⎭(5) ()()cos cos 2n n

ax b a ax b n π⎛⎫+=++⋅⎡⎤ ⎪⎣⎦⎝

⎭ (6)

()

()

()

1

1!

1n n n

n a n ax b ax b +⋅⎛⎫=- ⎪+⎝⎭

+ (7)

()()

()

()()

1

1!

ln 1n n n n

a n ax

b ax b -⋅-+=-⎡⎤⎣⎦

+

五、微分公式与微分运算法则

⑴()0

d c = ⑵

()1d x x dx

μμμ-= ⑶

()sin cos d x xdx

=

⑷()cos sin d x xdx

=- ⑸

()2tan sec d x xdx = ⑹

()2cot csc d x xdx

=-

⑺

()sec sec tan d x x xdx =⋅ ⑻

()csc csc cot d x x xdx

=-⋅

⑼

()x

x d e

e dx =

⑽

()ln x

x

d a

a

adx

= ⑾

()1ln d x dx x =

⑿()1log ln x

a d dx x a = ⒀(

)arcsin d x = ⒁(

)arccos d x =

⒂

()21arctan 1d x dx x =

+ ⒃()2

1

arccot 1d x dx x =-+

六、微分运算法则 ⑴

()d u v du dv

±=± ⑵

()d cu cdu

=

⑶()d uv vdu udv =+ ⑷2

u vdu udv

d v v -⎛⎫= ⎪⎝⎭

七、基本积分公式

⑴kdx kx c =+⎰ ⑵11x x dx c μμ

μ+=++⎰ ⑶ln dx x c x =+⎰

⑷ln x

x

a a dx c a =+⎰ ⑸x x e dx e c =+⎰ ⑹cos sin xdx x c =+⎰

⑺sin cos xdx x c =-+⎰ ⑻221

sec tan cos dx xdx x c x ==+⎰⎰ ⑼2

21csc cot sin xdx x c x ==-+⎰⎰ ⑽21arctan 1dx x c x =++⎰

⑾

arcsin x c

=+

八、补充积分公式

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 =-+⎰

22

11arctan x

dx c a x a a =++⎰

22

11ln 2x a

dx c x a a x a -=+-+⎰