微积分公式大全

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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

x1

Dx sin-1 ()=

22aax

x

cos-1 ()=

axatan-1 ()=2

aax2xcot-1 ()=

a

 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 dx = x sin-1 x+1x2+C  cos-1 x dx = x cos-1 x-1x2+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+x21|+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

x

sinh-1 ()= ln (x+a2x2) xR

ax

cosh-1 ()=ln (x+x2a2) x≧1

ax1ax

tanh-1 ()=ln () |x| <1

axa2a

1xa-1xcoth ()=ln () |x| >1 -1-12

xaa2a csc x dx = x csc x+ ln |x+x1|+C

xa

sec-1 ()=

axx2a2csc-1 (x/a)= Dx sinh x = cosh x

cosh x = sinh x tanh x = sech2 x coth x = -csch2 x

 sinh x dx = cosh x + C  cosh x dx = sinh x + C  tanh x dx = ln | cosh x |+ C  coth x dx = ln | sinh x | + C

x11x2

sech()=ln(+)0≦x≦1

xax2

-1

x11x2

csch ()=ln(+) |x| >0 2

axx

duv = udv + vdu

-1

 duv = uv =  udv +  vdu → udv = uv -  vdu cos2θ-sin2θ=cos2θ cos2θ+ sin2θ=1 cosh2θ-sinh2θ=1

cosh2θ+sinh2θ=cosh2θ

sech x = -sech x tanh x  sech x dx = -2tan-1 (e-x) + C csch x = -csch x coth x 1ex

 csch x dx = 2 ln || + C

2x

1e


x

Dx sinh()=

a

-1

1ax1xa

22

2

 sinh-1 x dx = x sinh-1 x-1x2+ C  cosh-1 x dx = x cosh-1 x-x21+ C

sin 3θ=3sinθ-4sin3θ cos3θ=4cos3θ-3cosθ →sin3θ= ¼ (3sinθ-sin3θ) →cos3θ=¼(3cosθ+cos3θ)

x

cosh()=

a

-1

2

xatanh-1()= 2

aax2xcoth()=

a

-1

 tanh-1 x dx = x tanh-1 x+ ½ ln | 1-x2|+ C ejxejxejxejx

sin x = cos x =

2 coth-1 x dx = x coth-1 x- ½ ln | 1-x2|+ C 2j

exexexex

sinh x = cosh x =

22abc

正弦定理:= ==2R

sinsinsin

 sech-1 x dx = x sech-1 x- sin-1 x + C

 csch-1 x dx = x csch-1 x+ sinh-1 x + C

xa γ sech-1()=

22a axax R b csch-1(x/a)=

axax

2

2



β

α

c

余弦定理: a2=b2+c2-2bc cosα

b2=a2+c2-2ac cosβ

c2=a2+b2-2ab cosγ



sin (α±β)=sin α cos β ± cos α sin β cos (α±β)=cos α cos β sin α sin β 2 sin α cos β = sin (α+β) + sin (α-β) 2 cos α sin β = sin (α+β) - sin (α-β) 2 cos α cos β = cos (α-β) + cos (α+β) 2 sin α sin β = cos (α-β) - cos (α+β)

x2x3xn

e=1+x+++…++ …

2!3!n!

x

sin α + sin β = 2 sin ½(α+β) cos ½(α-β) sin α - sin β = 2 cos ½(α+β) sin ½(α-β)

cos α + cos β = 2 cos ½(α+β) cos ½(α-β) cos α - cos β = -2 sin ½(α+β) sin ½(α-β) tan (α±β)=

tantancotcot

, cot (α±β)=

tantancotcot

1= n

i1n

n

x3x5x7(1)nx2n1

sin x = x-+-+…++ …

3!5!7!(2n1)!x2x4x6(1)nx2n

cos x = 1-+-+…++ …

2!4!6!(2n)!x2x3x4(1)nxn1

ln (1+x) = x-+-+…++ …

234(n1)!

i= ½n (n+1)

i1n

i2=

i1n

1

n (n+1)(2n+1) 6

i

i1

3

= [½n (n+1)]2





2

x3x5x7(1)nx2n1

tan x = x-+-+…++ …

357(2n1)

-1

r

Γ(x) = tx-1e-t dt = 2t2x-1etdt =

0

0





0

1

(ln)x-1 dt t

r(r1)2r(r1)(r2)31

m-1n-1(1+x)=1+rx+x+x+… -1β(m, n) =x(1-x) dx=22sin2m-1x cos2n-1x dx

002!3!

=

希腊字母

大写

Α Β Γ Δ

小写 α β γ δ

读音 alpha beta gamma delta

大写 Ι Κ Λ Μ

小写 ι κ λ μ

读音 iota kappa lambda mu

大写 Ρ Σ Τ Υ





0

xm1

dx mn

(1x)

小写 ρ σ, ς τ υ

读音 rho sigma tau upsilon

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