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1、微积分及三角函数基本公式Dx sin x=cos xcos x = -sin xtan x = sec2 xcot x = -csc2 xsec x = sec x tan xcsc x = -csc x cot x1xDx sin-1 ()=22aa xxcos-1 ()=ax atan-1 ()=2aa 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
2、 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 sec-1 x dx = x sec-1 x- ln |x+x21|+Csin-1(-x) = -sin-1 xcos-1(-x) = - cos-1 xtan-1(-x) = -tan-1 xcot-1(-x) = - cot-1 xsec-1(-x) = - sec-
3、1 xcsc-1(-x) = - csc-1 xxsinh-1 ()= ln (x+a2 x2) xRaxcosh-1 ()=ln (x+x2a2) x1ax1a xtanh-1 ()=ln () |x| 1cotha2ax a csc-1 x dx = x csc-1 x+ ln |x+x21|+Cx asec-1 ()=axx2 a2csc-1 (x/a)=Dx sinh x = cosh xcosh x = sinh xtanh x = sech2 xcoth x = -csch2 xsech x = -sech x tanh xcsch x = -csch x coth xDx sin
4、h-1(x)=a1a x1x a222sech-1(x11 x2)=ln(+)0x12axx 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 |2x11 xcsch-1 ()=ln(+) |x| 02axxduv = udv + vdu duv = uv =udv +vdu1 ex1e2x| + Cudv = uv -vducos2-sin2
5、=cos2cos2+ sin2=1cosh2-sinh2=1cosh2+sinh2=cosh2sin 3=3sin-4sin3cos3=4cos3-3cossin3= (3sin-sin3)cos3=(3cos+cos3)ejx e jxejx e jxsin x =cos x =2 j2 sinh-1 x dx = x sinh-1 x-1 x2+ C cosh-1 x dx = x cosh-1 x-x21+ C tanh-1 x dx = x tanh-1 x+ ln | 1-x2|+Cxcosh-1()=a2tanh-1(x a)=2aa x2-1 x dx = x coth-1 x-
6、 ln | 1-x2|+ C cothxex exex ex-1coth ()=sinh x =cosh x =-1-1-1a sech x dx = x sech x- sin x + C22 axbca csch-1 x dx = x csch-1 x+ sinh-1 x + C-1sech ()=正弦定理:=2R22asinsinsinx a x2=b2+c2-2bc cosa餘弦定理: aRb acsch-1(x/a)=b2=a2+c2-2ac cos22xa xcc2=a2+b2-2ab cossin ()=sin cos cos sin cos ()=cos cos sin sin
7、 2 sin cos = sin (+) + sin (-)2 cos sin = sin (+) - sin (-)2 cos cos = cos (-) + cos (+)2 sin sin = cos (-) - cos (+)ex=1+x+x2x3xn+ 2!3!n!sin + sin = 2 sin (+) cos (-)sin - sin = 2 cos (+) sin (-)cos + cos = 2 cos (+) cos (-)cos - cos = -2 sin (+) sin (-)tan tan cotcottan ()=, cot ()= tantancot cot1
8、= ni1nn(1)nx2n1x3x5x7sin x = x-+-+ (2n 1)!3!5!7!(1)nx2nx2x4x6cos x = 1-+-+ (2n)!2!4!6!(1)nxn1x2x3x4ln (1+x) = x-+-+ (n 1)!234i= n (n+1)i1i2=i1nn1n (n+1)(2n+1)6ii13= n (n+1)2x-1e-t dt = 2t2x-1etdt =t0012tan-1 x = x-(1)nx2n1x3x5x7+-+ (2n 1)357(x) =01(ln )x-1 dtt(1+x)r=1+rx+r(r 1)2r(r 1)(r 2)3x +x + -1
9、x12!3!(m, n) =x0m-1(1-x)n-1 dx=220sin2m-1x cos2n-1x dx=希臘字母希臘字母 (Greek Alphabets)大寫倒數關係: sincsc=1; tancot=1; cossec=1商數關係: tan=小寫讀音alphabetagammadeltaepsilonzetaetatheta大寫小寫讀音iotakappalambdamunuxiomicronpi大寫0xm1dxmn(1 x)小寫讀音rho,sigmatauupsilonphikhipsiomegasincos; cot=cossin平方關係: cos2+ sin2=1; tan2+
10、 1= sec2; 1+ cot2= csc2順位高;順位高 d 順位低 ;順位低0* =1* =10= 0*=0000=e0();0=e0;1=e0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)算術平均數(Arithmetic mean)中位數(Median)眾數(Mode)幾何平均數(Geometric mean)調和平均數(Harmonic mean)X X1 X2. Xnn取排序後中間的那位數字次數出現最多的數值G nX1X2.XnH 11111(.)n x1x2xni平均差(Average Deviatoin)變異數(Variance)|
11、 X1n X |n X)2(X1ninor(X1ni X)2n1標準差(Standard Deviation)(X1ni X)2n分配DiscreteUniformContinuousUniformBernoulliBinomialNegativeBinomialMultinomial機率函數 f(x)期望值 E(x)or(X1ni X)2n 1變異數 V(x)動差母函數m(t)1 et(1ent)tn1eebteat(b a)t1n1b a1(n+1)21(a+b)212(n +1)121(b-a)212pxq1-x(x=0, 1)nx n-xxp q k x 1k xp qxf(x1, x
12、2, ,xm-1)=n!xxxp11p22.pmmx1!x2!.xm!pnpkqppqnpqkqp2q+pet(q+ pet)npk(1 qet)knpinpi(1-pi)三項(p1et1+p2et2+ p3)nGeometricpqx-11pq2ppet1 qetHypergeometrickN kxn x Nnexx! k nN N n k nN 1 NPoissonNormalBetaGamma2e(et1)12e1 x2()2et2t2121x1(1 x)1B(,)(1)()2(x)1ex()x121tExponentChi-Squared2=f(2) e=1 n22n2222tE(
13、)=n(2)n122V( )=2ne2(12t)n2Weibull1ex1 121 22 11 1 1 000 000 000 000 000 000 000 000 1024 yotta Y1 000 000 000 000 000 000 000 1021 zetta Z1 000 000 000 000 000 000 1018 exa E1 000 000 000 000 000 1015 peta P1 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 ki
14、lo 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（或寫作厘 ） ，百分之一10-3 milli m 毫，千分之一001 10-6 micro微，百萬分之一000 001 10-9 nano n 奈，十億分之一000 000 001 10-12 pico p 皮，兆分之一000 000 000 001 10-15 femto f 飛（或作費 ） ，千兆分之一000 000 000 000 001 10-18 atto a 阿000 000 000 000 000 001 10-21 zepto z000 000 000 000 000 000 001 10-24 yocto y
