1、微積分公式Dx sin x=cos xcos x = -sin xtan x = sec2 xcot x = -csc2 xsec x = sec x tan xcsc x = -csc x cot x 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 | + Csin-1(-x) = -sin-1 xcos-1(-x) = -
2、cos-1 xtan-1(-x) = -tan-1 xcot-1(-x) = - cot-1 xsec-1(-x) = - sec-1 xcsc-1(-x) = - csc-1 xDx sin-1 ( )= a21cos-1 ( )=tan-1 ( )=ax2cot-1 ( )=sec-1 ( )=ax2acsc-1 (x/a)= sin-1 x dx = x sin-1 x+ +C21 cos-1 x dx = x cos-1 x- +Cx 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
3、x dx = x sec-1 x- ln |x+ |+C1x csc-1 x dx = x csc-1 x+ ln |x+ |+C2sinh-1 ( )= ln (x+ ) x Ra2cosh-1 ( )=ln (x+ ) x1xatanh-1 ( )= ln ( ) |x| 1axaxsech-1( )=ln( + )0x112csch-1 ( )=ln( + ) |x| 0ax2Dx sinh x = cosh xcosh x = sinh xtanh x = sech2 xcoth x = -csch2 xsech x = -sech x tanh xcsch x = -csch x c
4、oth x 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 | | + Cx21duv = udv + vdu duv = uv = udv + vdu udv = uv - vducos2-sin 2=cos2cos2+ sin 2 =1cosh2 -sinh2=1cosh2 +sinh2=cosh2Dx sinh-1( )= a21xco
5、sh-1( )= 2tanh-1( )= ax2coth-1( )=sech-1( )= ax2xcsch-1(x/a)= 2 sinh-1 x dx = x sinh-1 x- + C21x cosh-1 x dx = x cosh-1 x- + C tanh-1 x dx = x tanh-1 x+ ln | 1-x2|+ C coth-1 x dx = x coth-1 x- ln | 1-x2|+ C sech-1 x dx = x sech-1 x- sin-1 x + C csch-1 x dx = x csch-1 x+ sinh-1 x + Csin 3 =3sin-4sin
6、3cos3 =4cos3-3cossin 3 = (3sin-sin3)cos 3 =(3cos+cos3 )sin x = cos x = jejj22jxjesinh x = cosh x = xx正弦定理: = = =2Rsinaibsinc餘弦定理: a 2=b2+c2-2bc cosb2=a2+c2-2ac 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 (
7、-) + cos (+)2 sin sin = cos (-) - cos (+)sin + sin = 2 sin (+) cos (-)sin - sin = 2 cos (+) sin (-)cos + cos = 2 cos (+) cos (-)cos - cos = -2 sin (+) sin (-)tan ()= , cot ()=tantcottex=1+x+ + + + !23x!nsin x = x- + - + + !5!7)!12(xncos x = 1- + - + + !24x!6)!(n= ni1= n (n+1)i1= n (n+1)(2n+1)i126a b
8、cRln (1+x) = x- + - + + 2x34)!1(nxtan-1 x = x- + - + + 57)2(1+x)r =1+rx+ x2+ x3+ -1x1!)1(!r= n (n+1)2ni13(x) = x-1e-t dt = 2 2x-1 dt = x-1 dt0t02e0)1(ln(m, n ) = m-1(1-x)n-1 dx=2 2m-1x cos2n-1x dx 1xsi= dx01)(n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 iota rho beta kappa , sigma gamma l
9、ambda tau delta mu upsilon epsilon nu phi zeta xi khi eta omicron psi theta pi omega倒數關係: sincsc =1; tancot=1; cos sec=1商數關係: tan= ; cot= cosinsinco平方關係: cos 2+ sin 2=1; tan 2+ 1= sec2; 1+ cot 2= csc 2; 順位高 d 順位低 ; 順 位 低順 位 高0* = * = = 0* = 101= ; = ; = 0)(e0e0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;
10、 三角(雙曲)算術平均數(Arithmetic mean) nXXn.21中位數(Median) 取排序後中間的那位數字眾數(Mode) 次數出現最多的數值幾何平均數(Geometric mean) nnXG.21調和平均數(Harmonic mean) )1.1(2nxxnH平均差(Average Deviatoin) nXi|1變異數(Variance)or i21)(1)(2nXni標準差(Standard Deviation)or nXni21)()(21i分配 機率函數 f(x) 期望值 E(x) 變異數 V(x) 動差母函數m(t)Discrete Uniform n1(n+1)2
11、1(n2+1)1tne1Continuous Uniform ab(a+b) (b-a)2 tabtt)(Bernoulli pxq1-x(x=0, 1) p pq q+petBinomial pxqn-xnp npq (q+ pet)nNegative Binomial pkqx1pkq2pkqktqep)1(Multinomial f(x1, x2, , xm-1)= mxn.!.2121 npi npi(1-pi)三項(p1et1+ p2et2+ p3)nGeometric pqx-1 p12pqtq1Hypergeometric nNxkn Nkn1NkPoisson !xe )1(teNormal 2)(1 2 2 1 tteBeta 11)(),(xB2)(1(Gamma xe1)( 2tExponent x121tChi-Squared 2 =f( 2)= 212(ennE( 2)=n V( 2)=2n 2)1(ntWeibull xe 111122
