微积分公式大全

cos2θ+ sin2θ=1
tanh x
csch x dx = 2 ln | 1 ex | + C cosh2θ-sinh2θ=1
csch x = -csch x
1 e2x
cosh2θ+sinh2θ=cosh2θ
coth x
Dx sinh-1 x =
1
a
a2 x2
sin 3θ=3sinθ-4sin3θ sinh-1 x dx = x sinh-1 x- 1 x2 + C cos3θ=4cos3θ-3cosθ
2 cos α sin β = sin α+β - sin α-β cos α - cos β = -2 sin α+β sin α-β
2 cos α cos β = cos α-β + cos α+β tan α±β= tan tan , cot α±
2 sin α sin β = cos α-β - cos α+β
1 b-a2
12
pq npq
q+pet q+ petn
npi1-pi
三项 p1et1+ p2et2+
pn 3
N n n k
N 1 N
λ
σ2
Vχ2=2n
10-1 deci d 分,十分之一 10-2 centi c 厘或写作「厘」,百分之一 10-3 milli m 毫,千分之一 001 10-6 micro 微,百万分之一 000 001 10-9 nano n 奈,十亿分之一 000 000 001 10-12 pico p 皮,兆分之一 000 000 000 001 10-15 femto f 飞或作「费」,千兆分之一 000 000 000 000 001 10-18 atto a 阿 000 000 000 000 000 001 10-21 zepto z 000 000 000 000 000 000 001 10-24 yocto y
tan tan
β= cot cot cot cot
ex=1+x+ x2 + x3 +…+ xn + …
2! 3!
n!
sin x = x- x3 + x5 - x7 +…+ (1)n x2n1 + …
3! 5! 7!
(2n 1)!
cos x = 1- x2 + x4 - x6 +…+ (1)n x2n + …
Δ δ delta Μ μ mu Υ υ
Ε
ε epsilon Ν
ν
nu
Φ
φ
Ζ ζ zeta Ξ ξ
xi
Χχ
Η η eta Ο ο omicron Ψ ψ
Θ θ theta Π π
pi
Ωω
倒数关系: sinθcscθ=1; tanθcotθ=1; cosθsecθ=1
读音 rho sigma tau upsilon phi khi psi omega
duv = uv = udv + vdu
tanh x = sech2 x tanh x dx = ln | cosh x |+ C → udv = uv - vdu
coth x = -csch2 x coth x dx = ln | sinh x | + C cos2θ-sin2θ=cos2θ
sech x = -sech x sech x dx = -2tan-1 e-x + C
2
csch-1x/a= x
a a2 x2
csch-1 x dx = x cschβ-1 x+ sinh-1 x e x ex
+C
2
Ra
γ
b
α
c
正弦定理: a = b = c =2R sin sin sin
余弦定理: a2=b2+c2-2bc cosα b2=a2+c2-2ac cosβ
c2=a2+b2-2ab cosγ
商数关系: tanθ= sin ; cotθ= cos
c os
s in
平方关系: cos2θ+ sin2θ=1; tan2θ+ 1= sec2θ; 1+ cot2θ= csc2θ
順位高 ; 顺位高 d 顺位低 ;
順位低
0 = 1 = = 01 = 0
00
顺位一: 对数; 反三角反双曲 顺位二: 多项函数; 幂函数
2! 4! 6!
(2n)!
ln 1+x = x- x2 + x3 - x4 +…+ (1)n x n1 + …
2 34
(n 1)!
tan-1 x = x- x3 + x5 - x7 +…+ (1)n x2n1 + …
357
(2n 1)
n
1= n
i 1
n
i = n n+1
i 1
n
i2 =
1
n n+12n+1
tan-1-x = -tan-1 x
cot x = -csc2 x cot x dx = ln |sin x | + C
cot-1-x = - cot-1 x
sec x = sec x tan x sec x dx = ln |sec x + tan x | + sec-1-x = - sec-1 x
变异数 Variance
n
n
(Xi X)2
(Xi X )2
1
or 1
n
n 1
标准差 Standard Deviation
n
(Xi X )2
1
or
n
n
(Xi X)2
1
n 1
分配
机率函数 fx
期望值 Ex
变异数 Vx
Discrete Uniform
1 n+1
2
1 n2+1
12
动差母函 数 mt
Continuous Uniform Bernoulli
i 1
6
n
i3 = n n+12
i 1
Γx =
t
e x-1 -t
dt
=
2
td = t e 2x-1 t2
(ln 1) x-1 dt
0
0
0t
1+xr =1+rx+ r(r 1) 2!
x2+ r(r 1)(r 2) x3+… 3!
-1<x<1 βm,
n
=
1
x
m-11-xn-1
0
dx=2
微积分公式
Dx sin x=cos x cos x = -sin x
sin x dx = -cos x + C cos x dx = sin x + C
sin-1-x = -sin-1 x cos-1-x = - cos-1 x
tan x = sec2 x
tan x dx = ln |sec x | + C
cosh-1 x = a
1 x2 a2
→sin3θ= 3sinθ-sin3θ cosh-1 x dx = x cosh-1 x- x2 1 + C →cos3θ=3cosθ+cos3θ
tanh-1 x = a a a2 x2
tanh-1 x dx = x tanh-1 x+ ln | sin x = e jx e jx cos x =
1-x2|+ C
2j
coth-1 x = a
coth-1 x dx = x coth-1 x- ln | e jx e jx
1-x2|+ C
2
sech-1 x =
Hale Waihona Puke asech-1 x dx = x sech-1 x- sin-1 x + sinh x = e x ex cosh x =
a x a2 x2 C
Binomial
Negative Binomial Multinomial
Geometric
Hypergeomet ric Poisson
pxq1-xx=0, 1
n x
p qx n-x
k
x x
1
pkqx
fx1, x2, …, xm-1=
n! x1! x 2 !... x m !
p x1 1
p2
x2
...
sin α±β=sin α cos β ± cos α sin β sin α + sin β = 2 sin α+β cos α-β
cos α±β=cos α cos β sin α sin β sin α - sin β = 2 cos α+β sin α-β
2 sin α cos β = sin α+β + sin α-β cos α + cos β = 2 cos α+β cos α-β
a
cos-1 x dx = x cos-1 x- 1 x2 +C tan-1 x dx = x tan-1 x-ln 1+x2+C cosh-1
x =ln x+ a
x2 a2
x≧1
tan-1 x = a a a2 x2
cot-1 x = a
cot-1 x dx = x cot-1 x+ln 1+x2+C tanh-1 x = 1 ln a x |x| <1
sech-1 x =ln 1 + 1 x2 0≦x≦1
ax
x2
csch-1 x =ln 1 + 1 x2 |x| >0
ax
x2
Dx sinh x = cosh x sinh x dx = cosh x + C
duv = udv + vdu
cosh x = sinh x cosh x dx = sinh x + C
pm
xm
pqx-1
Normal Beta Gamma Exponent Chi-Squared χ2
Weibull
=fχ2
=
1
(
2
)
n 2
1
e
2
2
n
2
n 2
2
1 a+b
2
p np
npi
n k
N
λ
μ
Eχ2=n
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 1 000 000 000 000 1012 tera T 兆 1 000 000 000 109 giga G 十亿 1 000 000 106 mega M 百万 1 000 103 kilo K 千 100 102 hecto H 百 10 101 deca D 十
csc x = -csc x cot C
csc-1-x = - csc-1 x
x
csc x dx = ln |csc x – cot x |
+C
Dx sin-1 x = a
cos-1 x = a
1 a2 x2
sin-1 x dx = x sin-1 x+ 1 x2 +C sinh-1 x = ln x+ a2 x2 x R
a 2a a x
sec-1 x dx = x sec-1 x- ln
coth-1 x = 1 ln x a |x| >1
|x+ x2 1 |+C
a 2a x a
sec-1 x = a a x x2 a2
csc-1 x/a=
csc-1 x dx = x csc-1 x+ ln |x+ x2 1 |+C
0 0 = e0() ; 0 = e 0 ; 1 = e0
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
2 sin x 2m-1
0
cos2n-1x
dx
= xm1 dx
0 (1 x)mn
希腊字母 Greek Alphabets
大写 小写 读音 大写 小写 读音 大写 小写
Α α alpha Ι ι iota Ρ ρ
Β β beta Κ κ kappa Σ σ,
Γ γ gamma Λ λ lambda Τ τ