1、微積分公式Dx sin x=cos xcos x = -sin x tan x = sec2 x cot x = -csc2 xsec x = sec x tan x csc x = -csc x cot xDx sin-1 ( x )=1x2aa2cos-1 ( x )=a21ax2tan-1 ( x )=a2ax2acot-1 ( x )=a2a2ax-1xasec ()=a x x2 a2csc-1 ( x )=aa2ax x2sin x dx = -cos x + Ccos x dx = sin x + Ctan x dx = ln |sec x | + Ccot x dx = ln

2、|sin x | + Csec x dx = ln |sec x + tan x | + C csc x dx = ln |csc x cot x | + Csin-1 x dx = x sin-1 x+1x 2+Ccos-1 x dx = x cos-1 x-1x2+Ctan-1 x dx = x tan-1 x-? ln (1+x2 )+Ccot-1 x dx = x cot-1 x+? ln (1+x 2)+Csec-1 x dx = x sec-1 x- ln|x+x 21 |+Ccsc-1 x dx = x csc-1 x+ ln|x+x 21 |+Csin-1(-x) = -sin

3、 -1 xcos-1(-x) =- cos-1 xtan-1(-x) = -tan-1 xcot-1(-x) =- cot-1 xsec-1(-x) =- sec-1 xcsc-1(-x) = - csc-1 xsinh-1 ( x )= ln (x+a2x 2 ) x Racosh-1 ( x )=ln (x+x2a 2 ) x 1atanh-1 ( x )=1ln ( ax ) |x| <1a2aaxcoth-1 ( x)=1ln ( xa ) |x| >1a2axa-1x)=ln(11x2sech (+x 2)0x 1ax-1x)=ln(11x2csch(+x 2) |x|

4、>0axDx sinh x = cosh xsinh x dx = cosh x + Ccosh x = sinh xcosh x dx = sinh x + C2tanh x dx = ln | cosh x |+ Ctanh x = sech xcoth x = -csch2 xcoth x dx = ln | sinh x | + Csech x = -sech x tanh x-1-x) + Csech x dx = -2tan (ecsch x = -csch x coth xcsch x dx = 2 ln | 1ex| + C1e 2 xDx sinh-1 ( x )=a

5、21x2-1-12asinhx dx = x sinhx-1x+ Ccosh-1( x )=1a 2-1-12ax2coshx dx = x coshx-x1 + Ctanh-1( x )=atanh-1 x dx = x tanh-1 x+?ln | 1-x2|+a2x2Cacoth-1( x )=acoth-1 x dx = x coth-1 x-?ln | 1-x2|+a22Caxsech-1( x )=asech-1 x dx = x sech-1 x- sin-1 x + Caxa 2x2csch-1 x dx = x csch-1 x+ sinh-1 x + Ccsch-1( x

6、)=aaxx 2Rbaa 2cduv = udv + vduduv = uv =udv +vduudv = uv -vdu22cos -sin =cos222cos + sin =122cosh -sinh =122cosh +sinh =cosh23sin 3=3sin-4sin 3cos3 =4cos -3cos sin3 = ? (3sin -sin3) cos3=?(3cos +cos3)sin x =ejxe jxcos x =e jxe jx2 j2sinh x =exe xcosh x = exe x22ab=c=2R正弦定理:=sinsinsin余弦定理 : a222=b +c

7、 -2bc cosb2=a2 +c2-2ac cos c2=a2+b2-2ab cossin (±)=sin cos ± cos sin cos (±)=cos cos sin sin 2 sin cos = sin (+) + sin (-)2 cos sin = sin (+) - sin (-)2 cos cos = cos (-) + cos (+)2 sin sin = cos (-) - cos (+)xx2x3xne =1+x+ +2!3!x 7n!x3x5n2 n 1sin x = x-+-+ + (1)x1)!+3!5!7!( 2nx 2x 4x

8、 6+ +(1)nx2ncos x = 1-+4!-+2!6!(2n)!x2x 3x4nxn1ln (1+x) = x-2+-+ +(1)+34(n1)!tan-1 x = x- x3+ x5- x7+ + (1) n x 2n1+357(2n1)sin + sin = 2 sin ? (+) cos ? (-)sin - sin = 2 cos ? (+) sin ? (-)cos + cos = 2 cos ? (+) cos ? (-)cos - cos = -2 sin ? (+) sin ? (-)tan (±)= tantan, cot (±)=cotcotta

9、ntancotcotn1 = ni 1ni = ? n (n+1)i 1n1i 2 =n (n+1)(2n+1)i 16ni 3 = ? n (n+1)2i 1(x) =tx-1 -tdt = 2t2x-1et2(ln1 x-1dtedt =)000t(1+x)r =1+rx+ r (r1)x2+ r ( r1)( r2)x3+ -1<x<1 (m, n) =x m-1 (1-x) n-1 dx=22 sin 2m-1x cos2n -1x dx12!3!00=xm 10 (1x)m n dx希臘字母 (Greek Alphabets)大寫小寫讀音大寫小寫讀音大寫小寫讀音alpha

10、iotarhobetakappa, ?sigmagammalambdataudeltamuupsilonepsilonnuphizetaxikhietaomicronpsithetapiomega倒數(shù)關(guān)系 : sincsc=1; tancot=1; cossec=1sincos商數(shù)關(guān)系 : tan=; cot =cossin平方關(guān)系 : cos2 + sin2=1; tan2 + 1= sec2; 1+ cot2 = csc2順位高順位高 d順位低 ;順位低1*=100* = 0*=0000 = e0() ;0 = e0; 1 = e0順位一 : 對(duì)數(shù) ; 反三角 (反雙曲)順位二 : 多項(xiàng)函

11、數(shù) ; 冪函數(shù)順位三 : 指數(shù) ; 三角(雙曲)算術(shù)平均數(shù) (Arithmetic mean)XX1X 2. X nn中位數(shù) (Median)取排序后中間的那位數(shù)字眾數(shù) (Mode)次數(shù)出現(xiàn)最多的數(shù)值幾何平均數(shù) (Geometric mean)GnX1X 2. X nH1調(diào)和平均數(shù) (Harmonic mean)1111)(x1x2.nxn平均差 (Average Deviatoin)n| X iX |1nnX )2nX )2變異數(shù) (Variance)( X i( X i1or1nn 1nX )2nX ) 2標(biāo)準(zhǔn)差 (Standard Deviation)( X i( X i1or1nn1動(dòng)

12、差母函數(shù)分配DiscreteUniformContinuousUniformBernoulliBinomialNegativeBinomialMultinomialGeometricHypergeometric機(jī)率函數(shù) f(x)1n1bapxq1-x(x=0, 1)n pxqn-xxk x1 pkqxxf(x1, x2, , xm-1 )=n!p1x1 p2x2 . pm xmx1 ! x2!. xm !pqx-1kNkxnxNn期望值 E(x)變異數(shù) V(x)1(n+1)1 (n2+1)2121(a+b)1(b-a)2212ppqnpnpqkqkqpp 2npinpi(1-pi)1qpp 2

13、n kNn nkNN1Nm(t)t nt 1 e (1 e )ebteat(ba)tq+pet(q+ pe t)npk(1qet ) k三項(xiàng)t1(p1e +t2np2e + p3)tpe1qetPoissonexx!1 xe(et1)NormalBetaGammaExponentChi-Squared 21e()2221x1 (1x)1B( ,)( x)1 e x()ex=f(2)1n21=2 ) 2n (e 2n 2 222(1)()22112E(2)=nV(2)=2nt12 t2e2ttn(12t ) 2Weibull1xe112121211-1 -23691215182124-3-6-9-12000 100 100tera100-十15Tpeta億100-兆18exaP10-21Ezetta-24yottaZ Y10010十千百giga G百 萬(wàn)1