微积分及三角函数公式

微积分及三角函数基本公式

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

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

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

axx2-1

xasec-1 ()=

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

xDx sinh()=

a-1 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

2axxduv = 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)

1a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

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

ejxejxejxejxsin x = cos x =

2j2 coth-1 x dx = x coth-1 x- ½ ln | 1-x2|+ 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

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 sin α + sin β = 2 sin ½(α+β) cos ½(α-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 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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)357-1i= ½n (n+1)

i1i2=

i1nn1 n (n+1)(2n+1)

6ii13= [½n (n+1)]2

Γ(x) =

t0x-1-te dt = 2t022x-1tedt =

01(ln)x-1 dt

t(1+x)r

=1+rx+1r(r1)2r(r1)(r2)3x+x+… -1

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

002!3!=

希臘字母 (Greek Alphabets)

大寫

Α

Β

Γ

Δ

Ε

Ζ

Η

Θ

小寫 讀音

alpha

beta

gamma

delta

epsilon

zeta

eta

theta

大寫

Ι

Κ

Λ

Μ

Ν

Ξ

Ο

Π

小寫 讀音

iota

kappa

lambda

mu

nu

xi

omicron

pi

大寫

Ρ

Σ

Τ

Υ

Φ

Χ

Ψ

Ω

0xm1dx

(1x)mn小寫 讀音

α

β

γ

δ

ε

ζ

η

θ

ι

κ

λ

μ

ν

ξ

ο

π

rho

ρ

σ, ς

sigma

tau

τ

υ

upsilon

phi

φ

khi

χ

psi

ψ

ω

omega

倒數關係: sinθcscθ=1; tanθcotθ=1; cosθsecθ=1

sincos; cotθ=

cossin平方關係: cos2θ+ sin2θ=1; tan2θ+ 1= sec2θ; 1+ cot2θ= csc2θ

商數關係: tanθ=

順位高;  順位高d 順位低 ;

順位低1100* = * = = 0* =

0000 =

e0() ;

0 =

e0 ;

1 =

e0

算術平均數(Arithmetic mean)

XX2...XnX1

n順位一: 對數; 反三角(反雙曲)

順位二: 多項函數; 冪函數

順位三: 指數; 三角(雙曲)

中位數(Median)

眾數(Mode)

幾何平均數(Geometric mean)

調与平均數(Harmonic mean)

取排序後中間的那位數字

次數出現最多的數值

GnX1X2...Xn

H1

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

變異數(Variance)

|X1nX|n

X)2(X1nin or

(X1niX)2n1

標準差(Standard Deviation)

(X1niX)2n分配

Discrete

Uniform

Continuous

Uniform

Bernoulli

Binomial

Negative

Binomial

Multinomial

機率函數f(x)

1

n or

(X1niX)2

變異數V(x)

12(n+1)

12n1期望值E(x)

1(n+1)

2動差母函數m(t)

1et(1ent)

tn1eebteat

(ba)t1

ba1(a+b)

21(b-a)2

12pxq1-x(x=0, 1)

nxn-xxpq

kx1kxpq

xf(x1, x2, …, xm-1)=

pmm

x1!x2!...xm!p

np

kq

ppq

npq

kqp2q+pet

(q+ pet)n

pk(1qet)k三項

(p1et1+

p2et2+ p3)n

npi

1

pkn

Nnpi(1-pi)

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

λ λ

e(et1)

Normal

Beta

Gamma

12e1x2 ()2

μ σ2

e

 t2 t212

1x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

22



tE(χ)=n

(2)n122V(χ)=2n

e22(12t)n2

Weibull

1e

x

121

221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 十億

1 000 000 106 mega M 百萬

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(或寫作「厘」),百分之一

0、001 10-3 milli m 毫,千分之一

0、000 001 10-6 micro ? 微,百萬分之一

0、000 000 001 10-9 nano n 奈,十億分之一

0、000 000 000 001 10-12 pico p 皮,兆分之一

0、000 000 000 000 001 10-15 femto f 飛(或作「費」),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