微積分轉化公式

(1)微積分的基本公式共有四大公式：

1、牛頓-萊布尼茨公式，又稱為微積分基本公式

2、格林公式，把封閉的曲線積分化為區域內的二重積分，它是平面向量場散度的二重積分

3、高斯公式，把曲面積分化為區域內的三重積分，它是平面向量場散度的三重積分

4、斯托克斯公式，與旋度有關

(2)微積分常用公式：

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 ()=

cos-1 ()=

tan-1 ()=

cot-1 ()=

sec-1 ()=

csc-1 (x/a)=

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

cos-1 x dx = x cos-1 x-+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+|+C

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

sinh-1 ()= ln (x+) xR

cosh-1 ()=ln (x+) x≥1

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

sech-1()=ln(+)0≤x≤1

csch-1 ()=ln(+) |x| >0

Dx sinh x = cosh x

cosh x = sinh x

tanh x = sech2 x

coth x = -csch2 x

sech x = -sech x tanh x

csch x = -csch x coth 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

sech x dx = -2tan-1 (e-x) + C

csch x dx = 2 ln || + C

duv = udv + vdu

duv = uv = udv + vdu

→ udv = uv - vdu

cos2θ-sin2θ=cos2θ

cos2θ+ sin2θ=1

cosh2θ-sinh2θ=1

cosh2θ+sinh2θ=cosh2θ

Dx sinh-1()=

cosh-1()=

tanh-1()=

coth-1()=

sech-1()=

csch-1(x/a)=

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

cosh-1 x dx = x cosh-1 x-+ C

tanh-1 x dx = x tanh-1 x+ ln | 1-x2|+ C

coth-1 x dx = x coth-1 x- ln | 1-x2|+ C

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

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

sin 3θ=3sinθ-4sin3θ

cos3θ=4cos3θ-3cosθ

→sin3θ= (3sinθ-sin3θ)

→cos3θ= (3cosθ+cos3θ)

sin x = cos x =

sinh x = cosh x =

正弦定理:= ==2R

餘弦定理: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)

sin α + sin β = 2 sin (α+β) cos (α-β)

sin α - sin β = 2 cos (α+β) sin (α-β)

cos α + cos β = 2 cos (α+β) cos (α-β)

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

tan (α±β)=，cot (α±β)=

ex=1+x+++…++ …

sin x = x-+-+…++ …

cos x = 1-+-+++

ln (1+x) = x-+-+++

tan-1 x = x-+-+++

(1+x)r =1+rx+x2+x3+ -1= n

= n (n+1)

= n (n+1)(2n+1)

= [ n (n+1)]2

Γ(x) = x-1e-t dt = 22x-1dt = x-1 dt

β(m，n) =m-1(1-x)n-1 dx=22m-1x cos2n-1x dx = dx