微積分基本公式

1、微積分公式Dx sin x=cos xcos x = -s in xtan x = sec2 xcot x = -csc2 x sec x = sec x tan xcsc x = -csc x cot xsin x dx = -cos x + C cos x dx = sin x + C tan x dx = In |sec x | + C cot x dx = In |sin x | + C sec x dx = In |sec x + tan x | + C csc x dx = In |csc x -cot x | + C.-1 -1sin (-x) = -sin x cos-1(-x

2、) =- cos-1 xtan-1 (-x) = -tan -1 x cot-1 (-x) =- cot-1 xsec-1(-x) =- sec-1 xcsc (-x) = - csc x1 x atan ()=22a a x1 x acot ( 一)= 一22a a xsec-1x()=aax x2a21 xacsc ()= a 忖a2Dx si nh x = cosh xcosh x = sinh x tanh x = sech x coth x = -csch x sech x = -sech x tanh x csch x = -csch x coth x1 1 . 2sin_ x d

3、x = x sin ' x+ 1 x +Ccos-1tan-1cot-1x dx = x cos -1x dx = x tan -1x dx = x cot -1x- . 1x2 +Cx-?ln (1+x 2)+C2x+?ln (1+x 2)+C11f 2“sec x dx = x sec x- In |x+ . x 1 |+Ccsc-1 x dx = x csc -1 x+ In |x+ x 1 |+Csinh x dx = cosh x + C cosh x dx = sinh x + Ctanh x dx = In | cosh x |+ Ccoth x dx = In | si

4、nh x | + C1xsech x dx = -2tan - (e-) + Cx| + C 2x e1 ecsch x dx = 2 In | 1 x22sinh 1 ( )= In (x+ a x ) x R a-1 x22cosh ( )=ln (x+ x a ) x = 1 a1x1axtanh 1 (_ )=In ()兇 <1a2aax1x1xacoth 1 ()=In ()兇 >1a2axa2“X 11 xsech-1()=ln(+、2)0 全x全 1a x . x1 x 1 i11x2csch1 ( )=ln( + 2) |x| >0a x : xduv = u

5、dv + vduduv = uv = udv + vdu t udv = uv - vdu cos2 9 -sin2 0 =cos2 0 cos2 0 + sin2 0 =1 cosh2 0 -sinh2 0 =1 cosh2 0 +sinh2 0 =cosh2 0Dx sinh-1( x)=a1 xcosh ()=a2 2a x12 2x a11i,2sinh1 x dx = x si nh 1 x- 1 x + C11I 2cosh x dx = x cosh x-詁 x 1 + Ctan h-1(x)=atan h-1coth-1sin 3 0 =3sin 0 -4sin3 9cos3

6、0 =4cos3 0 -3cos 0f sin3 0 = ? (3sin 0 -sin3 0 )f cos3 0 =?(3cos 0 +cos3 0 )coth-1()=a-isech-1csch-11 xsech ()=aaX、a2 x21 x csch ()= aax . a2 x2sin ( a±®=s in a cos p 士 cos a si n pcos ( a±p=cos a cos p sin asin p2 sina cos p= sin ( a+ p) + sin ( a- p)2 cosasin p= sin ( a+ p) - sin (

7、a- p2 cosa cos p= cos ( a p + COS ( a+ p2 sina sin p= cos ( a- p) - COS ( a+ p)23nx x x x e =1+x+ + + + + 2!3 n!sin x = x-+3!5!7!cos x = 1-2 x4 x+6 x357x x x-+2!4!6!12x dx = x tanh x+ ? In | 1-x |+ C -1 x- ? In | 1-x 2|+ C1x- sin x + Cejx e jxsin x =ejxe jxx dx = x cothx dx = x sechx dx = x cschn 2n

8、 1(1) x(2n1)!(1)nx2n(2n)!+ + 234n n 1x x x ( 1) xIn (1+x) = x- + - + +234(n 1)!n 2n 1(1) x+ 3571 x x xtan x = x- + - + + ;+ 357(2n1)-isinsincoscos2jcos x =xxe esinh x =cosh x =正弦定理：sinc=2Rsin sin余弦定理：a2=b2+c2-2bc cos a.2 2 2b =a +c -2ac cos p2 2 .2c =a +b -2ab cos 丫a+ sin p= 2 sin ?( a+ p) cos ?( a-

9、 pa- sin p= 2 cos ?( a+ p) sin ?( a pa + cos p= 2 cos ?( a+ p COS ?( a p)a- cos p= -2 sin ?( a+ p sin ?( a- ptan tancot cottan ( a士p)= , cot ( a士p)=tan tancot coti = ?n (n + 1)(1+x)r=1 + rx+r(r 1)x2+ r(r 1)(r2)x3+ -1<x<12!3!1-n (n+1)(2 n+1)6=?n (n+1)2r (x)=0tp (m, n)=x-1e-t dt = 2 ot 2x-11m-1n

10、-1 Iqx (1-x) dx=2m 1x0m n dx0 (1 x)t2dt =(ln -)x-1 dtt2 sin2m-1x cos2n-1x dx =大寫小寫、-H W 讀曰大寫小寫、-H W 讀曰大寫小寫、一K & 讀曰AaalphaI1iotapPrhoBpbetaKKkappa藝O , ?sigmarYgammaA入lambdaTTtauASdeltaMamuYUupsil on希臘字母(Greek Alphabets)seedE£epsil onNVnu©phiZZ :zeta口ExiXXkhiHnetao0omicro n屮psi00thetannp

11、iQ3omega倒數(shù)關(guān)系:sin 0 csc0 =1; tan 0 cot 0 =1; cos 0 sec 0 =1cossin+ 1= sec2 0 ; 1+ cot2 0 = csc2 0sin商數(shù)關(guān)系：tan 0 =； cot 0 =cos平方關(guān)系:cos2 0 + sin2 0 =1; tan2 0順位高；順位高d順位低；順位低0*10=0* _ =000彳0=e ； 1= e順位一：對數(shù)；反三角(反雙曲)順位二：多項函數(shù)；幕函數(shù)00 = e0()；0算術(shù)平均數(shù)(Arithmetic mean)順位三：指數(shù)；三角(雙曲)中位數(shù)(Median)取排序后中間的那位數(shù)字眾數(shù)(Mode)次數(shù)岀

12、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)(Varianee)n(Xi X)21orn2(Xi X)1n(Xi X)2or1標(biāo)準(zhǔn)差(Standard Deviation)n(Xi X)21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 Uniform2( n+1)丄(n 2+1)12Con ti nuousUn iform2(a+b)(b-a)212、丿Ber noullipXq1-x(x=0, 1)ppqq+petBi nomialn pxqn-xXnpn

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