微积分的基本公式1

- 1 - 微积分公式表

导数 函数 积分

幂函数系 0 kC kx

1x x—kx 11kxk)1(k

21x x1 xln

211x xarctan

221ax axaarctan1

221ax axaxaln21

211x xarcsin

221xa axarcsin

221xa )ln(22xax

221ax 22lnaxx

指数函数系 指数函数 xe xe xe

aaxln xa aaxln

对数函数 x1 xln xxxln

axln1 xalog axxxalnlog - 2 - 导数 函数 积分

三角函数系 三角函数 xcos xsin xcos

xsin xcos xsin

x2sec xtan xcosln

x2csc xcot xsinln

xxtansec xsec xxtansecln

)42tan(lnx

xxcotcsc xcsc xxcotcscln2tanlnx

x2sec xtan

x2csc xcot

xxtansec xsec

xxcotcsc xcsc

反三角函数 211x xarcsin 21arcsinxxx

211x xarccos 21arcsinxxx

211x xarctan )1ln(21arctan2xxx

211x xarccot )1ln(21arctan2xxx

微积分公式

Dx sin x=cos x

cos x = -sin x

tan x = sec2 x

cot x = -csc2 x

sec x = sec x tan x

csc x = -csc x cot x  sin x dx = -cos x + C

 cos x dx = sin x + C

 tan x dx = ln |sec x | + C

 cot x dx = ln |sin x | + C

 sec x dx = ln |sec x + tan x | + C

 csc x dx = ln |csc x – cot x | + C sin-1(-x) = -sin-1 x

cos-1(-x) =  - cos-1 x

tan-1(-x) = -tan-1 x

cot-1(-x) =  - cot-1 x

sec-1(-x) =  - sec-1 x

csc-1(-x) = - csc-1 x

Dx sin-1 (ax)=221ax

cos-1 (ax)=221ax

tan-1 (ax)=22aax

cot-1 (ax)=22aax

sec-1 (ax)=22axxa

csc-1 (ax)=22axxa  sin-1 x dx = x sin-1 x+21x+C

 cos-1 x dx = x cos-1 x-21x+C

 tan-1 x dx = x tan-1 x-?ln (1+x2)+C

 cot-1 x dx = x cot-1 x+?ln (1+x2)+C

 sec-1 x dx = x sec-1 x- ln |x+12x|+C

 csc-1 x dx = x csc-1 x+ ln |x+12x|+C sinh-1 (ax)= ln (x+22xa) xR

cosh-1 (ax)=ln (x+22ax) x≧1

微积分公式

Dx sin x=cos x

cos x = -sin x

tan x = sec2 x

cot x = -csc2 x

sec x = sec x tan x

csc x = -csc x cot

x  sin x dx = -cos x + C

 cos x dx = sin x + C

 tan x dx = ln |sec x | + C

 cot x dx = ln |sin x | + C

 sec x dx = ln |sec x + tan x |

+ C

 csc x dx = ln |csc x – cot x |

+ C sin-1(-x) = -sin-1 x

cos-1(-x) =  - cos-1 x

tan-1(-x) = -tan-1 x

cot-1(-x) =  - cot-1 x

sec-1(-x) =  - sec-1 x

csc-1(-x) = - csc-1 x

Dx sin-1 (ax)=221ax

cos-1 (ax)=221ax

tan-1 (ax)=22aax

cot-1 (ax)=22aax

sec-1 (ax)=22axxa

csc-1 (ax)=22axxa  sin-1 x dx = x sin-1 x+21x+C

 cos-1 x dx = x cos-1 x-21x+C

 tan-1 x dx = x tan-1 x-?ln (1+x2)+C

 cot-1 x dx = x cot-1 x+?ln (1+x2)+C

 sec-1 x dx = x sec-1 x- ln

|x+12x|+C

 csc-1 x dx = x csc-1 x+ ln

|x+12x|+C sinh-1 (ax)= ln (x+22xa)

xR

cosh-1 (ax)=ln (x+22ax) x≧1

微积分公式

Dx sin x=cos x

cos x = -sin x

tan x = sec2 x

cot x = -csc2 x

sec x = sec x tan x

csc x = -csc x cot x sin x dx = -cos x + C

cos x dx = sin x + C

tan x dx = ln |sec x | + C

cot x dx = ln |sin x | + C

sec x dx = ln |sec x + tan x | +

C

csc x dx = ln |csc x – cot x | +

C sin-1(-x) = -sin-1 x

cos-1(-x) = - cos-1 x

tan-1(-x) = -tan-1 x

cot-1(-x) = - cot-1 x

sec-1(-x) = - sec-1 x

csc-1(-x) = - csc-1 x

Dx sin-1 (ax)=221ax

cos-1 (ax)=221ax

tan-1 (ax)=22aax

cot-1 (ax)=22aax

sec-1 (ax)=22axxa

csc-1 (ax)=22axxa sin-1 x dx = x sin-1 x+21x+C

cos-1 x dx = x cos-1 x-21x+C

tan-1 x dx = x tan-1 x-½ln (1+x2)+C

cot-1 x dx = x cot-1 x+½ln (1+x2)+C

sec-1 x dx = x sec-1 x- ln

|x+12x|+C

csc-1 x dx = x csc-1 x+ ln

|x+12x|+C

sinh-1 (ax)= ln (x+22xa) xR

cosh-1 (ax)=ln (x+22ax) x≧1