2022年微积分公式大全.docx

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1、精选学习资料 - - - - - - - - - 名师精编 优秀资料微積分公式Dx sin x=cos x sin x dx = -cos x + C sin-1-x = -sin-1 x cos x = -sin x cos-1-x = - cos-1 x cos x dx = sin x + C tan x = sec 2 x tan-1-x = -tan-1 x tan x dx = ln |sec x | + C cot x = -csc 2 x cot-1-x = - cot-1 x cot x dx = ln |sin x | + C sec x = sec x tan x sec

2、-1-x = - sec-1 x sec x dx = ln |sec x + tan x | + C csc x = -csc x cot x csc-1-x = - csc-1 x csc x dx = ln |csc x cot x | + C Dx sin-1 x = aa21x2sinh-1 x = ln x+ aa2x2 xR sin-1 x dx = x sin-1 x+1x2+C cos-1 x dx = x cos-1 x-1x2+C cosh-1 x =ln x+ ax2a2 x1 cos-1 x = a tan-1 x dx = x tan-1 x-. ln 1+x2+C

3、 tan-1 x = aa2a2tanh-1 x = a1 ln 2 aax |x| 1 csc-1 x dx = x csc-1 x+ ln |x+x21|+C xasec-1 x = axx2aa2sech-1x =ln a1 + x1xx20x12 sinh x dx = cosh x + C csch-1 x =ln a1 + x1xx2 |x| 0 csc-1 x/a= 2Dx sinh x = cosh x duv = udv + vducosh x = sinh x duv = uv = udv + vdu cosh x dx = sinh x + C tanh x = sech

4、 2 x tanh x dx = ln | cosh x |+ C udv = uv - vducoth x = -csch 2 x cos 2 -sin 2 =cos2 coth x dx = ln | sinh x | + C sech x = -sech x tanh x sech x dx = -2tan-1 e-x + C cos 2 + sin 2 =1 csch x = -csch x coth x cosh 2 -sinh 2 =1 csch x dx = 2 ln |1exx| + C Dx sinh-1x = aa21x22cosh 2 +sinh 2 =cosh21esi

5、n 3 =3sin -4sin 3 sinh-1 x dx = x sinh-1 x-1x2+ C cos3 =4cos 3 -3cos cosh-1 x dx = x cosh-1 x-x21+ C sin 3 = . 3sin -sin3 cosh-1x = ax21a2cos 3 =.3cos +cos3 tanh-1 x dx = x tanh-1 x+ .ln | 1-x2|+ C sin x = ejx2ejxcos x = ejx2ejxtanh-1x = aa2a2 coth-1 x dx = x coth-1 x- .ln | 1-x2|+ C xjcoth-1x = a s

6、ech-1 x dx = x sech-1 x- sin-1 x + C sinh x = ex2excosh x = ex2ex csch-1 x dx = x csch-1 x+ sinh-1 x + C sech-1x = axa2ax2R a b 正弦定理 :a= b=c=2R sinsinsinc 餘弦定理 : a 2=b 2+c 2-2bc coscsch-1x/a=xa2ax2b 2=a 2+c 2-2ac cosc 2=a 2+b2-2ab cos名师归纳总结 第 1 页,共 4 页- - - - - - -精选学习资料 - - - - - - - - - 名师精编 优秀资料s

7、in =sin cos cos sin sin + sin = 2 sin . + cos . - cos =cos cos sin sin sin - sin = 2 cos . + sin . - 2 sin cos = sin + + sin - 2 cos sin = sin + - sin - cos + cos = 2 cos . + cos . - cos - cos = -2 sin . + sin . - 2 cos cos = cos - + cos + tan = tan tan , cot = cot cot2 sin sin = cos - - cos + tan t

8、an cot cot2 3 n ne x=1+x+ x + x + + x + 1= n.2 .3 n . i 13 5 7 n 2 n 1 nsin x = x-x + x -x+ + 1 x + i = . n n+1 .3 .5 .7 2 n 1 . i 12 4 6 n 2 n ncos x = 1-x+ x-x+ + 1 x + i 2 = 1 n n+12n+1 .2 .4 .6 2 n . i 1 62 3 4 n n 1 nln 1+x = x-x+ x-x + + 1 x + i 3 = . n n+1 22 3 4 n 1 . i 13 5 7 n 2 n 1tan-1 x

9、 = x-x +3 x -5 x7 + + 2 1n x1 + x = 0t x-1e-t dt = 2 0t 2x-1e 2tdt = 0 ln 1t x-1 dt1+x r =1+rx+ r r.2 1 x 2+ r r 1.3 r 2 x 3+ -1x1 m, n = 0x 1m-11-x n-1 dx=2 0 2sin 2m-1x cos 2n-1x dx m 1= 0 1 xx 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 iota rho beta kappa , . sigma gamma lambda t

10、au delta mu upsilon epsilon nu phi zeta xi khi eta omicron psi theta pi omega 倒數關係 : sin csc =1; tan cot =1; cos sec =1 商數關係 : tan = sin; cot = cos第 2 页,共 4 页cossin平方關係 : cos 2 + sin 2 =1; tan 2 + 1= sec 2 ; 1+ cot 2 = csc 2順位高; 順位高 d 順位低 ; 順位低0* = 1* = = 0*1= 0e0順位一 : 對數 ; 反三角 反雙曲 順位二 : 多項函數 ; 冪函數0

11、00 = e000= e0; ; 1= 順位三 : 指數 ; 三角雙曲 名师归纳总结 - - - - - - -精选学习资料 - - - - - - - - - 名师精编X优秀资料X2n.Xn算術平均數 Arithmetic mean X1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1幾何平均數 Geometric mean GnX1X2.Xn調和平均數 Harmonic mean H111.1平均差 Average Deviatoin nx 1x2nxn|XiX|1n安排變異數 Variance nX22nXiX22動差母函數Xi標準差 Standard D

12、eviation 1Xor 1n1nnnXiXiX機率函數 fx 1or 1n1n期望值 Ex 變異數 Vx mt Discrete x 1.1mxm1 n+1 21 n 122+1 1t e 1entUniform nn1etContinuous 11 a+b 21 b-a 212bt eat eUniform ba ba tpxq1-xx=0, 1 p pq Bernoulli q+pe tBinomial n p xxqn-xnp npq q+ petnNegative kx1 pkqxkqkqpkxp2Binomial p 1t qekfx1, x2, , xm-1= npinpi1-

13、pi三項Multinomial p1e t1+xn .xm.p 1x 1p2x 2.pGeometric 2.1qp2e t2+ p3 nt pepqx-1Hypergeometric pp21t qekNknk NNnnkxnxN1NNn名师归纳总结 - - - - - - -第 3 页,共 4 页精选学习资料 - - - - - - - - - 名师精编优秀资料11ette1 Poisson e.xxNormal 1e1x22e12t2222Beta B1x1 1x 11t,Gamma x1ex2Exponent ex11t2Chi-Squared2=f2 E2=nV2=2nn2 t2=1

14、2n2n1e222第 4 页,共 4 页n22Weibull 1ex11122121 000 000 000 000 000 000 000 000 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

15、千100 102 hecto H 百10 101 deca D 十0.1 10-1 deci d 分,非常之一0.01 10-2 centi c 厘（或寫作厘 ）,百分之一0.001 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 pico p 皮,兆分之一0.000 000 000 000 001 10-15 femto f 飛（或作費 ）,千兆分之一0.000 000 000 000 000 001 10-18 atto a 阿0.000 000 000 000 000 000 001 10-21 zepto z 0.000 000 000 000 000 000 000 001 10-24 yocto y名师归纳总结 - - - - - - -