微积分基本公式

Dx sin x=cos x 积分

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-ln (1+x2)+C cot-1 x dx = x cot-1 x+ln (1+x2)+C 公

sin-1(-x) = -sin-1 x 式

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x1x cos-1 ()= 22aaxcos-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xaxtan-1 ()=2 2aaxaxcot-1 ()=2 aax2 sec-1 x dx = x sec-1 x- ln |x+x21|+C 1xa-1xcoth ()=ln () |x| >1 a2axa csc-1 x dx = x csc-1 x+ ln |x+x21|+C 2sec-1 (x)=axx)=axaxaaxa222 x11x2sech()=ln(+)0≦x≦1axx2-1csc-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 csch x dx = 2 ln |x11x2csch ()=ln(+) |x| >0 2axx-1Dx 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 1ex1e2x| + C duv = udv + vdu duv = uv = udv + vdu → udv = uv - vdu cos2θ-sin2θ=cos2θ cos2θ+ sin2θ=1 cosh2θ-sinh2θ=1 cosh2θ+sinh2θ=cosh2θ 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-12sin 3θ=3sinθ-4sin3θ cos3θ=4cos3θ-3cosθ →sin3θ= (3sinθ-sin3θ) →cos3θ=(3cosθ+cos3θ) tanh-1 x dx = x tanh-1 x+ ln | 1-x2|+ C ejxejxejxejxsin x = cos x = -1-122j2 coth x dx = x coth x- ln | 1-x|+ C sech-1 x dx = x sech-1 x- sin-1 x + C aexexexex-1xcoth()=2 sinh x = cosh x = 2-1-1-1aax csch x dx = x csch x+ sinh x + C 22axbca γ sech-1()= 正弦定理:= ==2R22a asinsinsinxax R b 222α 余弦定理: a=b+c-2bc cosα β a-1x222csch()= c b=a+c-2ac cosβ 22axaxc2=a2+b2-2ab cosγ axtanh()= 2 2aaxsin (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 (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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= n (n+1) i1i2= i1nn1 n (n+1)(2n+1) 6ii13= [n (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大写 Α Β 小写 0xm1dx mn(1x)α β 读音 alpha beta 大写 Ι Κ 小写 ι κ 读音 iota kappa 大写 Ρ Σ 小写 ρ σ, sigma 读音 rho Γ Δ Ε Ζ Η Θ γ δ ε ζ η θ gamma delta epsilon zeta eta theta Λ Μ Ν Ξ Ο Π λ μ ν ξ ο π lambda mu nu xi omicron pi Τ Υ Φ Χ Ψ Ω τ υ φ χ ψ ω tau upsilon phi khi psi omega 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

順位高; 顺位高d 顺位低 ;

順位低0* = 101 * = = 0* = 0000 = e0() ; 0 = e0 ; 1 = e0 算术平均数(Arithmetic mean) 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) 中位数(Median) 众数(Mode) 几何平均数(Geometric mean) 调和平均数(Harmonic mean) 平均差(Average Deviatoin) 取排序后中间的那位数字 次数出现最多的数值 (X1niX)2变异数(Variance) n or X)2(X1nin1 (X1niX)2 or 标准差(Standard Deviation) n(X1niX)2 动差母函数m(t) n1分配 机率函数f(x) 期望值E(x) 变异数V(x) Discrete Uniform Continuous Uniform Bernoulli 1(n+1) 21(a+b) 212(n+1) 121(b-a)2 12 pxq1-x(x=0, 1) nxn-xxpq kx1kxpq xf(x1, x2, …, xm-1)= n!xxxp11p22...pmm x1!x2!...xm!p np pq npq q+pet (q+ pet)n Binomial Negative Binomial Multinomial npi kn Nnpi(1-pi) Nnkn N1N三项 (p1et1+ p2et2+ p3)n Geometric Hypergeometric Poisson Normal Beta Gamma Exponent pqx-1 λ μ =f(χ2) λ σ2 Chi-Squaredχ2 =1n22n2(2)n12e22 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

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

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

分之一