微积分基本公式

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

tanh-1 (ax)=a21ln (xaxa) |x| <1

coth-1 (ax)=a21ln (axax) |x| >1

sech-1(ax)=ln(x1+221xx)0≦x≦1csch-1 (ax)=ln(x1+221xx) |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 |xxee211| + 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(ax)= 221xa

cosh-1(ax)= 221ax

tanh-1(ax)= 22aax

coth-1(ax)=22aax

sech-1(ax)= 22xaxa

csch-1(ax)=22xaxa  sinh-1 x dx = x sinh-1 x-21x+ C

 cosh-1 x dx = x cosh-1 x-12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 = jeejxjx2 cos x = 2jxjxeesinh x = 2xxee cosh x = 2xxee正弦定理:sina= sinb=sinc=2R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) 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)=tantantantan, cot (α±β)=cotcotcotcot

ex=1+x+!22x+!33x+…+!nxn+ …

sin x = x-!33x+!55x-!77x+…+)!12()1(12nxnn+ …

cos x = 1-!22x+!44x-!66x+…+)!2()1(2nxnn+ …

ln (1+x) = x-22x+33x-44x+…+)!1()1(1nxnn+ …

tan-1 x = x-33x+55x-77x+…+)12()1(12nxnn+ …

(1+x)r =1+rx+!2)1(rrx2+!3)2)(1(rrrx3+… -1

nii1= ?n (n+1)

nii12= 61 n (n+1)(2n+1)

nii13= [?n (n+1)]2

Γ(x) = 0tx-1e-t dt = 20t2x-12tedt = 0)1(lntx-1 dt

β(m, n) =10xm-1(1-x)n-1 dx=220sin2m-1x cos2n-1x dx= 01)1(nmmxxdx

希腊字母 (Greek Alphabets)

大写 小写 读音 大写 小写 读音 大写 小写 读音

Α α alpha Ι ι iota Ρ ρ rho

Β β beta Κ κ kappa Σ σ, ? sigma a b

c α β γ

R Γ γ gamma Λ λ lambda Τ τ

tau

Δ δ delta Μ

μ mu Υ υ upsilon

Ε ε epsilon Ν ν nu Φ φ phi

Ζ ζ zeta Ξ ξ xi Χ χ khi

Η η eta Ο ο omicron Ψ ψ psi

Θ θ theta Π π pi Ω ω omega

倒数关系: sinθcscθ=1; tanθcotθ=1; cosθsecθ=1

商数关系: tanθ= cossin; cotθ= sincos

平方关系: cos2θ+ sin2θ=1; tan2θ+ 1= sec2θ; 1+ cot2θ= csc2θ

順位低順位高;  顺位高d 顺位低 ;

0* = 1 * =  = 0*01 = 00

00 = )(0e ; 0 = 0e ; 1 = 0e 顺位一: 对数; 反三角(反双曲)

顺位二: 多项函数; 幂函数

顺位三: 指数; 三角(双曲)

算术平均数(Arithmetic mean)

中位数(Median) 取排序后中间的那位数字

众数(Mode) 次数出现最多的数值

几何平均数(Geometric mean)

调和平均数(Harmonic mean)

平均差(Average Deviatoin)

变异数(Variance) nXXni21)( or

1)(21nXXni

标准差(Standard Deviation) nXXni21)( or

1)(21nXXni

分配 机率函数f(x) 期望值E(x) 变异数V(x) 动差母函数m(t) Discrete

Uniform 21(n+1) 121(n2+1)

Continuous

Uniform 21(a+b) 121(b-a)2

Bernoulli pxq1-x(x=0, 1) p pq q+pet

Binomial xnpxqn-x np npq (q+

pet)n

Negative

Binomial xxk1pkqx

Multinomial f(x1, x2, …, xm-1)=

mxmxxmpppxxxn...!!...!!212121 npi npi(1-pi) 三项

(p1et1+

p2et2+

p3)n

Geometric pqx-1

Hypergeometric nNk 1NnNnNk

Poisson λ λ

Normal μ σ2

Beta

Gamma

Exponent

Chi-Squaredχ2 =f(χ2)

=212222)(221ennn E(χ2)=n V(χ2)=2n

Weibull

1 000 000 000 000 000 000 000 000 1024 yotta Y

1 000 000 000 000 000 000 000 1021 zetta Z

1 000 000 000 000 000 000 1018 exa E

1 000 000 000 000 000 1015 peta P