高数微积分基本公式大全

高等数学微积分公式大全

一、基本导数公式⑴()0c ′=⑵1

x x

µ

µµ−=⑶()sin cos x x

′=⑷()cos sin x x ′=−⑸()2

tan sec x x ′=⑹()2

cot 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()1ln x x

′=

⑿(

)1log ln

x

a

x a

′=⒀(

)arcsin x ′=

⒁(

)arccos x ′=⒂()2

1arctan 1x x ′=+⒃()2

1

arccot 1x x ′=−

+⒄()1x ′=

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、微分公式与微分运算法则

⑴()0

d c =⑵()1

d x

x

dx

µ

µµ−=⑶()sin cos d x xdx

=⑷()cos sin d x xdx =−⑸()2

tan sec d x xdx

=⑹()2

cot 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()1

ln d x dx x

=

⑿(

)1

log ln x

a

d dx x a

=⒀(

)arcsin d x =

⒁(

)arccos d x =⒂()2

1

arctan 1d x dx x

=+⒃()2

1

arccot 1d x dx x

=−

+四、微分运算法则⑴()d u v du dv ±=±⑵()d cu cdu =⑶()d uv vdu udv =+⑷2u vdu udv d v v −⎛⎞=⎜

⎟⎝⎠

五、基本积分公式⑴kdx kx c

=+∫

⑵1

1

x x dx c

µµ

µ+=++∫⑶

ln 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 =−+∫

⑻

2

21sec tan cos dx xdx x c x ==+∫∫⑼

2

21csc cot sin xdx x c

x ==−+∫∫⑽2

1arctan 1dx x c x

=++∫⑾

arcsin dx x c

=+六、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

−=+−+∫

arcsin x

c

a =+ln x c

=七、下列常用凑微分公式积分型

换元公式

()()()1

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

=+()()()

11f 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 =()()()1ln 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 =()()()2cot 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

=