微积分公式大全 (2)

2023年12月16日发(作者：省锡中初三数学试卷解析)

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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

1xDx sin-1 ()=

22aaxx cos-1 ()=

axatan-1 ()=2

2aaxxcot-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

xsinh-1 ()= ln (x+a2x2) xR

axcosh-1 ()=ln (x+x2a2) x≧1

ax1axtanh-1 ()=ln () |x| <1

a2aax1xa-1xcoth ()=ln () |x| >1

2-1-1a2axa csc x dx = x csc x+ ln |x+x1|+C

x11x2sech()=ln(+)0≦x≦1

2axx-1sec-1 (

xa)=

axx2a2csc-1 (x/a)=

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

1ex csch x dx = 2 ln || + C

2x1ex11x2csch ()=ln(+) |x| >0

axx2duv = udv + vdu

-1 duv = uv =  udv +  vdu

→ udv = uv -  vdu

cos2θ-sin2θ=cos2θ

cos2θ+ sin2θ=1

cosh2θ-sinh2θ=1

cosh2θ+sinh2θ=cosh2θ

sin 3θ=3sinθ-4sin3θ

cos3θ=4cos3θ-3cosθ

→sin3θ= ¼ (3sinθ-sin3θ)

→cos3θ=¼(3cosθ+cos3θ)

xDx sinh()=

a-11ax1xa222

 sinh-1 x dx = x sinh-1 x-1x2+ C

 cosh-1 x dx = x cosh-1 x-x21+ C

xcosh-1()=

a-12xatanh()=

2

2aax-1 tanh-1 x dx = x tanh-1 x+ ½ ln | 1-x2|+ C

ejxejxejxejxsin x = cos x =

-1-122j2 coth x dx = x coth x- ½ ln | 1-x|+ C

exexexexsinh x = cosh x =

22bca正弦定理:= ==2R

sinsinsin sech-1 x dx = x sech-1 x- sin-1 x + C

xcoth()=

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

ax

γ

sech-1()=

22a

axax

R b

csch-1(x/a)=\'.

axax22

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)

x2x3xne=1+x+++…++ …

2!3!n!xsin 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tancotcottan (α±β)=, cot (α±β)=

tantancotcot1= n

i1nn(1)nx2n1x3x5x7sin x = x-+-+…++ …

(2n1)!3!5!7!(1)nx2nx2x4x6cos x = 1-+-+…++ …

(2n)!2!4!6!(1)nxn1x2x3x4ln (1+x) = x-+-+…++ …

(n1)!234(1)nx2n1x3x5x7tan x = x-+-+…++ …

(2n1)357-1r

i= ½n (n+1)

i1i2=

i1nn1 n (n+1)(2n+1)

6ii13= [½n (n+1)]2

x-1-t2x-1ttte dt = 2edt =

002Γ(x) =

01(ln)x-1 dt

t1r(r1)2r(r1)(r2)3m-1n-1(1+x)=1+rx+x+x+… -1

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

002!3!=

希腊字母 (Greek Alphabets)

大写

Α

Β

Γ

Δ

Ε

Ζ

Η

Θ

小写

0xm1dx

mn(1x)α

β

γ

δ

ε

ζ

η

θ

读音

alpha

beta

gamma

delta

epsilon

zeta

eta

theta

大写

Ι

Κ

Λ

Μ

Ν

Ξ

Ο

Π

小写

ι

κ

λ

μ

ν

ξ

ο

π

读音

iota

kappa

lambda

mu

nu

xi

omicron

pi

大写

Ρ

Σ

Τ

Υ

Φ

Χ

Ψ

Ω

小写

ρ

σ, ς

sigma

tau

τ

υ

upsilon

phi

φ

khi

χ

psi

ψ

ω

omega

读音

rho

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

商数关系: tanθ=

sincos; cotθ=

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

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順位高;  顺位高d 顺位低 ;

順位低0* =

101 * = = 0* =

0000 =

e0() ;

0 =

e0 ;

1 =

e0

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)

众数(Mode)

几何平均数(Geometric mean)

调和平均数(Harmonic mean)

XX1X2...Xn

n取排序后中间的那位数字

次数出现最多的数值

GnX1X2...Xn

H1

1111(...)nx1x2xni平均差(Average Deviatoin)

变异数(Variance)

|X1nX|n

X)2(X1nin or

(X1niX)2n1

标准差(Standard Deviation)

(X1niX)2n

分配

Discrete

Uniform

Continuous

Uniform

Bernoulli

Binomial

Negative

Binomial

Multinomial

机率函数f(x) 期望值E(x)

or

(X1niX)2

n1变异数V(x) 动差母函数m(t)

1et(1ent)

tn1eebteat

(ba)t1

n1

ba1(n+1)

21(a+b)

212(n+1)

121(b-a)2

12pxq1-x(x=0, 1)

nxn-xxpq

kx1kxpq

xf(x1, x2, …, xm-1)=

p

np

kq

ppq

npq

kqp2q+pet

(q+ pet)n

pk(1qet)k三项

(p1et1+

npi

npi(1-pi)

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pmm

x1!x2!...xm!p2et2+ p3)n

1

pkn

Nq

2pNnkn

N1NGeometric

Hypergeometric

pqx-1

kNkxnx

Nnex

x!pet

1qet

Poisson

Normal

Beta

Gamma

λ

μ

λ

σ2

e(et1)

12e1x2 ()2e

 t2 t2121x1(1x)1

B(,)(1)()2

(x)1ex

()x

1



21

tExponent

Chi-Squaredχ2 =f(χ2)

e=1n22n2

22

t

n2E(χ)=n

(2)n122V(χ)=2n

2e2(12t)

Weibull

1e

x1



12122111



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

0.1 10-1 deci d 分，十分之一

0.01 10-2 centi c 厘（或写作「厘」），百分之一

0.001 10-3 milli m 毫，千分之一

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0.000 001 10-6 micro ? 微，百万分之一

0.000 000 001 10-9 nano n 奈，十亿分之一

0.000 000 000 001 10-12 pico p 皮，兆分之一

0.000 000 000 000 001 10-15 femto f 飞（或作「费」），千兆分之一

0.000 000 000 000 000 001 10-18 atto a 阿

0.000 000 000 000 000 000 001 10-21 zepto z

0.000 000 000 000 000 000 000 001 10-24 yocto y

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