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1、高数微积分公式大全微一積分公式 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 x x1Dx sin-1 = 22aa-xx cos-1 = axatan-1 =2 aa+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
2、 | + 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 sec-1 x dx = x sec-1 x- ln |x+x2-1|+C sin-1(-x) = -sin-1 x cos-1(-x) = p - cos-1 x tan-1(-x) = -tan-1 x cot-1(-x) = p - cot-1 x se
3、c-1(-x) = p - sec-1 x csc-1(-x) = - csc-1 x xsinh-1 = ln (x+a2+x2) xR axcosh-1 =ln (x+x2-a2) x1 ax1a+xtanh-1 =ln |x| 1 -1-12a2ax-a csc x dx = x csc x+ ln |x+x-1|+C x-11-x2sech=ln(+)0x1 2axx-1 xasec-1 = axx2-a2csc-1 (x/a)= Dx sinh x = cosh x cosh x = sinh x sinh x dx = cosh x + C cosh x dx = sinh x +
4、 C x11+x2csch =ln(+) |x| 0 2axxduv = udv + vdu -1 tanh x = sech2 x tanh x dx = ln | cosh x |+ C coth x = -csch2 x coth x dx = ln | sinh x | + C sech x = -sech x tanh x sech x dx = -2tan-1 (e-x) + C csch x = -csch x coth x 1+e-x csch x dx = 2 ln | + C -2x1-ex1Dx sinh-1= sinh-1 x dx = x sinh-1 x-1+x2+
5、 C 22aa+xxcosh-1= a-1 duv = uv = udv + vdu udv = uv - vdu cos2-sin2=cos2 cos2+ sin2=1 cosh2-sinh2=1 cosh2+sinh2=cosh2 sin 3=3sin-4sin3 cos3=4cos3-3cos sin3= (3sin-sin3) cos3=(3cos+cos3) 1x-a22 cosh-1 x dx = x cosh-1 x-x2-1+ C tanh-1 x dx = x tanh-1 x+ ln | 1-x2|+ C xatanh= 2 aa-x2xcoth= a-1ejx+e-jxe
6、jx-e-jxsin x = cos x = 22j coth-1 x dx = x coth-1 x- ln | 1-x2|+ C ex-e-xex+e-xsinh x = cosh x = 22abc正弦定理:= =2R sinasinbsing sech-1 x dx = x sech-1 x- sin-1 x + C csch-1 x dx = x csch-1 x+ sinh-1 x + C x-a sech-1= 22a axa-x R b csch-1(x/a)= -axa+x22 c 餘弦定理: a2=b2+c2-2bc cos b2=a2+c2-2ac cos c2=a2+b
7、2-2ab cos sin ()=sin cos cos sin cos ()=cos cos msin sin 2 sin cos = sin (+) + sin (-) 2 cos sin = sin (+) - sin (-) 2 cos cos = cos (-) + cos (+) 2 sin sin = cos (-) - cos (+) x2x3xne=1+x+ 2!3!n!xsin + sin = 2 sin (+) cos (-) sin - sin = 2 cos (+) sin (-) cos + cos = 2 cos (+) cos (-) cos - cos = -
8、2 sin (+) sin (-) tan ()=tanatanbmcotacotb, cot ()= mtanatanbcotacotb1= n i=1nnx3x5x7(-1)nx2n+1sin x = x-+-+ 3!5!7!(2n+1)!x2x4x6(-1)nx2ncos x = 1-+-+ 2!4!6!(2n)!x2x3x4(-1)nxn+1ln (1+x) = x-+-+ 234(n+1)!i= n (n+1) i=1ni2= i=1n1 n (n+1)(2n+1) 6ii=13= n (n+1)2 2x3x5x7(-1)nx2n+1tan x = x-+-+ 357(2n+1)-1
9、r (x) = tx-1e-t dt = 2t2x-1e-tdt = 0001(ln)x-1 dt tp1r(r-1)2r(r-1)(r-2)3m-1n-1(1+x)=1+rx+x+x+ -1x1 (m, n) =x(1-x) dx=22sin2m-1x cos2n-1x dx 002!3!= 希臘字母 (Greek Alphabets) 大寫 小寫 讀音 alpha beta gamma delta epsilon zeta eta theta 大寫 小寫 讀音 iota kappa lambda mu nu xi omicron pi 大寫 0xm-1dx m+n(1+x)小寫 讀音 rh
10、o , sigma tau upsilon phi khi psi omega 倒數關係: sincsc=1; tancot=1; cossec=1 商數關係: tan= sinqcosq; cot= cosqsinq平方關係: cos2+ sin2=1; tan2+ 1= sec2; 1+ cot2= csc2 順位高; 順位高d 順位低 ; 順位低1100* = * = = 0* = 00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) 00 = e0(-) ; 0 = e0 ; 1 = e0 算術平均數(Arithmetic mean) 中
11、位數(Median) 眾數(Mode) 幾何平均數(Geometric mean) 調和平均數(Harmonic mean) X1+X2+.+Xnn取排序後中間的那位數字 X=次數出現最多的數值 G=nX1X2.Xn H=11111(+.+)nx1x2xni平均差(Average Deviatoin) 變異數(Variance) |X1n-X|n-X)2(X1nin or (X1ni-X)2n-1標準差(Standard Deviation) (X1ni-X)2n分配 Discrete Uniform Continuous Uniform Bernoulli Binomial Negative
12、 Binomial Multinomial 機率函數f(x) 1 n1 b-a or (X1ni-X)2n-1期望值E(x) 1(n+1) 21(a+b) 2變異數V(x) 12(n+1) 121(b-a)2 12動差母函數m(t) 1et(1-ent) tn1-eebt-eat(b-a)tpxq1-x(x=0, 1) nxn-xxpq k+x-1kxpq xf(x1, x2, , xm-1)= n!xxxp11p22.pmm x1!x2!.xm!p np kq ppq npq kqp2q+pet (q+ pet)n pk(1-qet)k三項 (p1et1+ p2et2+ p3)n npi 1 pnpi(1-pi) q 2pN-nkn N-1NGeometric Hypergeometric pq kN-kxn-x Nnx-1pet 1-qetkn NPoisson Normal Beta Gamma e-llx x! 2e-l(et-1)12pse1x-m2- 2seaa+ba l1m t+s2 t2121xa-1(1-x)b-1 B(a,b)ab(a+b+1)(a+b)2l(lx)a-1e-lx G(a)a l21 2ll l-t-aExponent Chi-Squared2 =f(2) le=1nG22n
