微积分及三角函数公式

Dx sin x=cos x ? sin x dx = -cos x + C sin(-x) = -sin x cos x = -sin x ? cos x dx = sin x + C cos-1(-x) = ? - cos-1 x tan x = sec2 x ? tan x dx = ln |sec x | + C 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 | + C sec-1(-x) = ? - sec-1 x csc x = -csc x cot x ? csc x dx = ln |csc x – cot x | + csc-1(-x) = - csc-1 x C 1x-1-12Dx sin-1 ()= ? sin x dx = x sin x+1x+C sinh-1 (x)= ln (x+a2x2) xR aaa2x2-1-12? cos x dx = x cos x-1x+C xx-1-122 cos ()= cosh ()=ln (x+) x≧1 xa-1-12aa? tan x dx = x tan x-?ln (1+x)+C tan-1 (cot-1 (sec-1 (xa)=2 aax2x)= a-1-1? cot-1 x dx = x cot-1 x+?ln (1+x2)+C tanh-1 (x)=1ln (ax) |x| <1 a2aax-1-1? sec x dx = x sec x- ln x1xa-1coth ()=ln () |x| >1 2a2axa|x+x1|+C -1-1? csc x dx = x csc x+ ln xa)= axx2a2|x+x21|+C x11x2sech()=ln(+)0≦x2axx≦1 -1csc-1 (x/a)= Dx sinh x = cosh x ? sinh x dx = cosh x + C cosh x = sinh x ? cosh x dx = sinh x + C tanh x = sech2 x ? tanh x dx = ln | cosh x |+ C coth x = -csch2 x ? coth x dx = ln | sinh x | + C sech x = -sech x tanh ? sech x dx = -2tan-1 (e-x) + C x 1ex? csch x dx = 2 ln || + C 2x csch x = -csch x coth 1ex x11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 31xsin 3θ=3sinθ-4sinθ Dx sinh()= ? sinh-1 x dx = x sinh-1 x-1x2+ C acos3θ=4cos3θ-3cosθ a2x23 1x→sinθ= ? (3sinθ-sin3θ)-1cosh()= ? cosh-1 x dx = x cosh-1 x-x21+ C 22a→cos3θ=?(3cosθ+cos3θ) xa-1xatanh()= 2 aax2-1coth-1(sech-1(axax-1x)= ax)= a22 csch(x/a)=axax22 ? tanh-1 x dx = x tanh-1 x+ ? ln | 1-x2|+ ejxejxsin x = cos x = 2jC ? coth-1 x dx = x coth-1 x- ? ln | 1-x2|+ ejxejx C 2? sech-1 x dx = x sech-1 x- sin-1 x + C exexsinh x = cosh x = ? csch-1 x dx = x csch-1 x+ sinh-1 x + 2C exexβ 2γ a bca R b 正弦定理:= ==2R α sinsinsin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 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) 2 cos α sin β = sin (α+β) - sin (α-β) cos α - cos β = -2 sin ?(α+β) sin ?(α-β) tantan2 cos α cos β = cos (α-β) + cos (α+β) tan (α±β)=, cot (α±tantan2 sin α sin β = cos (α-β) - cos (α+β) cotcotβ)= cotcotx2x3xne=1+x+++…++ … 2!3!n!x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= ?n (n+1) i1ii1nn21= n (n+1)(2n+1) 6ii13= [?n (n+1)]2 x-1-t2x-11t2ettΓ(x) = e dt = 2dt = (ln)x-1 dt 000t(1+x)r =1+rx+1m-1r(r1)2r(r1)(r2)3n-1x+x+… -1大写 Α Β Γ Δ Ε Ζ Η Θ

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

sincos; cotθ= cossin小写 读音 alpha beta gamma delta epsilon zeta eta theta 大写 Ι Κ Λ Μ Ν Ξ Ο Π 小写 读音 iota kappa lambda mu nu xi omicron pi 大写 Ρ Σ Τ Υ Φ Χ Ψ Ω 小写 读音 α β γ δ ε ζ η θ ι κ λ μ ν ξ ο π rho ρ σ, ? sigma tau τ υ upsilon phi φ khi χ psi ψ ω omega 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*? = 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取排序後中间的那位数字 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

or (X1niX)2 n1分配 Discrete Uniform Continuous Uniform Bernoulli Binomial Negative Binomial Multinomial 机率函数f(x) 1 n1 ba期望值E(x) 1(n+1) 21(a+b) 2变异数V(x) 1(n2+1) 121(b-a)2 12动差母函数m(t) 1et(1ent) n1etebteat (ba)tpxq1-x(x=0, 1) nxn-xxpq kx1kxpq xp np kq ppq npq kqp2q+pet (q+ pe) pk(1qet)ktn f(x1, x2, …, xm-1)= n!xxxp11p22...pmm x1!x2!...xm!npi 1 pnpi(1-pi) q p2三项 (p1et1+ p2et2+ p3)n Geometric Hypergeometric pq kNkxnx Nnex x!x-1pet t1qe nk NNnkn N1N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 2222 t n2E(χ)=n (2)n122V(χ)=2n (12t) e 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

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