微積分基本公式

1、微積分公式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 xò sin x dx = -cos x + Cò cos x dx = sin x + Cò tan x dx = ln |sec x | + Cò cot x dx = ln |sin x | + Cò sec x dx = ln |sec x + tan x | + Cò csc x dx = ln |csc x cot

2、x | + Csin-1(-x) = -sin-1 xcos-1(-x) = p - cos-1 xtan-1(-x) = -tan-1 xcot-1(-x) = p - cot-1 xsec-1(-x) = p - sec-1 xcsc-1(-x) = - csc-1 xDx sin-1 ()= cos-1 ()=tan-1 ()=cot-1 ()=sec-1 ()=csc-1 ()=ò sin-1 x dx = x sin-1 x+Cò cos-1 x dx = x cos-1 x-+Cò tan-1 x dx = x tan-1 x-½ln (1+

3、x2)+Cò cot-1 x dx = x cot-1 x+½ln (1+x2)+Cò sec-1 x dx = x sec-1 x- ln |x+|+Cò csc-1 x dx = x csc-1 x+ ln |x+|+Csinh-1 ()= ln (x+) xRcosh-1 ()=ln (x+) x1tanh-1 ()=ln () |x| <1coth-1 ()=ln () |x| >1sech-1()=ln(+)0x1csch-1 ()=ln(+) |x| >0Dx sinh x = cosh x cosh x = sinh x

4、tanh x = sech2 x coth x = -csch2 x sech x = -sech x tanh x csch x = -csch x coth xò sinh x dx = cosh x + Cò cosh x dx = sinh x + Cò tanh x dx = ln | cosh x |+ Cò coth x dx = ln | sinh x | + Cò sech x dx = -2tan-1 (e-x) + Cò csch x dx = 2 ln | + Cduv = udv + vduò du

5、v = uv = ò udv + ò vduò udv = uv - ò vducos2-sin2=cos2cos2+ sin2=1cosh2-sinh2=1cosh2+sinh2=cosh2Dx sinh-1()= cosh-1()= tanh-1()= coth-1()=sech-1()= csch-1()=ò sinh-1 x dx = x sinh-1 x-+ Cò cosh-1 x dx = x cosh-1 x-+ Cò tanh-1 x dx = x tanh-1 x+ ½ ln | 1-x2|+ C

6、ò coth-1 x dx = x coth-1 x- ½ ln | 1-x2|+ Cò sech-1 x dx = x sech-1 x- sin-1 x + Cò csch-1 x dx = x csch-1 x+ sinh-1 x + CaRcbsin 3=3sin-4sin3cos3=4cos3-3cossin3= ¼ (3sin-sin3)cos3=¼(3cos+cos3)sin x = cos x = sinh x = cosh x = 正弦定理:= =2R余弦定理: a2=b2+c2-2bc cosb2=a2+c2-2a

7、c cosc2=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 (+)sin + sin = 2 sin ½(+) cos ½(-)sin - sin = 2 cos ½(+) sin ½(-)cos + cos = 2 co

8、s ½(+) cos ½(-)cos - cos = -2 sin ½(+) sin ½(-)tan (±)=, cot (±)=ex=1+x+ sin x = x-+-+ cos x = 1-+-+ ln (1+x) = x-+-+ tan-1 x = x-+-+ (1+x)r =1+rx+x2+x3+ -1<x<1= n= ½n (n+1)= n (n+1)(2n+1)= ½n (n+1)2(x) = x-1e-t dt = 22x-1dt = x-1 dt(m, n) =m-1(1-x)n-1 d

9、x=22m-1x cos2n-1x dx = 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alphaiotarhobetakappa, sigmagammalambdataudeltamuupsilonepsilonnuphizetaxikhietaomicronpsithetapiomega倒數(shù)關(guān)系: sincsc=1; tancot=1; cossec=1商數(shù)關(guān)系: tan= ; cot= 平方關(guān)系: cos2+ sin2=1; tan2+ 1= sec2; 1+ cot2= csc2; ò 順位高d 順位低 ; 0*¥ = *&

10、#165; = = 0* = = ; = ; = 順位一: 對數(shù); 反三角(反雙曲)順位二: 多項函數(shù); 冪函數(shù)順位三: 指數(shù); 三角(雙曲)算術(shù)平均數(shù)(Arithmetic mean)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)調(diào)和平均數(shù)(Harmonic mean)平均差(Average Deviatoin)變異數(shù)(Variance) or 標(biāo)準(zhǔn)差(Standard Deviation) or 分配機率函數(shù)f(x)期望值E(x)變異數(shù)V(x)動差母函數(shù)m(t)Discrete Uniform(n+1)(n2+1)C

11、ontinuous Uniform(a+b)(b-a)2Bernoullipxq1-x(x=0, 1)ppqq+petBinomialpxqn-xnpnpq(q+ pet)nNegative BinomialpkqxMultinomialf(x1, x2, , xm-1)=npinpi(1-pi)三項(p1et1+ p2et2+ p3)nGeometricpqx-1HypergeometricnnPoissonNormal2BetaGammaExponentChi-Squared2=f(2)=E(2)=nV(2)=2nWeibull1 000 000 000 000 000 000 000 0

12、00 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?qū)懽骼澹?，百分之?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