Wb/yue/微積分學/積分表

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Revision as of 17:02, 19 September 2023 by imported>Jonashtand (Created page with "{{Wb/yue/Calculus/Top Nav|導數表|}} == 運算法則 == * <math>\int c\cdot f(x)\mathrm{d}x=c\cdot\int f(x)\mathrm{d}x</math> * <math>\int\big(f(x)\pm g(x)\big)\mathrm{d}x=\int f(x)\mathrm{d}x\pm\int g(x)\mathrm{d}x</math> * <math>\int u\,dv=uv-\int v\,du</math> == 冪函數 == * <math>\int \mathrm{d}x=x+C</math> * <math>\int a\,\mathrm{d}x=ax+C</math> * <math>\int x^n\mathrm{d}x=\frac{x^{n+1}}{n+1}+C\qquad(n\ne-1)</math> * <math>\int\frac{\mathrm{d}x}{x}=\ln|x|+C<...")
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運算法則

  • cf(x)dx=cf(x)dx
  • (f(x)±g(x))dx=f(x)dx±g(x)dx
  • udv=uvvdu

冪函數

  • dx=x+C
  • adx=ax+C
  • xndx=xn+1n+1+C(n1)
  • dxx=ln|x|+C
  • dxax+b=ln|ax+b|a+C(a0)

三角函數

基本三角函數

  • sin(x)dx=cos(x)+C
  • cos(x)dx=sin(x)+C
  • tan(x)dx=ln|cos(x)|+C
  • sin2(x)dx=1cos(2x)2dx=x2sin(2x)4+C
  • cos2(x)dx=1+cos(2x)2dx=x2+sin(2x)4+C
  • tan2(x)dx=tan(x)x+C

倒數三角函數

  • sec(x)dx=ln|sec(x)+tan(x)|+C=ln|tan(x2+π4)|+C=2artanh(tan(x2))+C
  • csc(x)dx=ln|csc(x)+cot(x)|+C=ln|tan(x2)|+C
  • cot(x)dx=ln|sin(x)|+C
  • sec2(ax)dx=tan(ax)a+C
  • csc2(ax)dx=cot(ax)a+C
  • cot2(ax)dx=xcot(ax)a+C
  • sec(x)tan(x)dx=sec(x)+C
  • sec(x)csc(x)dx=ln|tan(x)|+C

降階公式

  • sinn(x)dx=sinn1(x)cos(x)n+n1nsinn2(x)dx+C(n>0)
  • cosn(x)dx=cosn1(x)sin(x)n+n1ncosn2(x)dx+C(n>0)
  • tann(x)dx=tann1(x)(n1)tann2(x)dx+C(n1)
  • secn(x)dx=secn1(x)sin(x)n1+n2n1secn2(x)dx+C(n1)
  • cscn(x)dx=cscn1(x)cos(x)n1+n2n1cscn2(x)dx+C(n1)
  • cotn(x)dx=cotn1(x)n1cotn2(x)dx+C(n1)
  • a2xnsin(ax)dx=nxn1sin(ax)axncos(ax)n(n1)xn2sin(ax)dx
  • a2xncos(ax)dx=axnsin(ax)+nxn1cos(ax)n(n1)xn2cos(ax)dx

顯形式

  • sinn(x)dx=cos(x)2F1(12,1n2;32;cos2(x))+C
  • cosn(x)dx=1n+1sgn(sin(x))cosn+1(x)2F1(12,n+12;n+32;cos2(x))+C(n1)
  • tann(x)dx=1n+1tann+1(x)2F1(1,n+12;n+32;tan2(x))+C(n1)
  • cscn(x)dx=cos(x)2F1(12,n+12;32;cos2(x))+C
  • secn(x)dx=sin(x)2F1(12,n+12;32;sin2(x))+C
  • cotn(x)dx=1n+1cotn+1(x)2F1(1,n+12;n+32;cot2(x))+C(n1)

其中2F1超幾何函數sgn符號函數

反三角函數

  • dx1x2=arcsin(x)+C
  • dxa2x2=arcsin(xa)+C(a0)
  • dx1+x2=arctan(x)+C
  • dxa2+x2=arctan(xa)a+C(a0)

指數同對數函數

  • exdx=ex+C
  • eaxdx=eaxa+C(a0)
  • axdx=axln(a)+C(a>0,a1)
  • ln(x)dx=xln(x)x+C
  • exsin(x)dx=ex2(sin(x)cos(x))+C
  • excos(x)dx=ex2(sin(x)+cos(x))+C

降階公式

  • xneaxdx=1axneaxnaxn1eaxdx

反三角函數

  • arcsin(x)dx=xarcsin(x)+1x2+C
  • arccos(x)dx=xarccos(x)1x2+C
  • arctan(x)dx=xarctan(x)12ln|1+x2|+C
  • arccsc(x)dx=xarccsc(x)+ln|x+x11x2|+C
  • arcsec(x)dx=xarcsec(x)ln|x+x11x2|+C
  • arccot(x)dx=xarccot(x)+12ln|1+x2|+C

雙曲函數

基本雙曲函數

  • sinh(x)dx=isin(ix)dx=cos(ix)+C=cosh(x)+C
  • cosh(x)dx=cos(ix)dx=isin(ix)+C=sinh(x)+C
  • tanh(x)dx=itan(ix)dx=log|cos(ix)|+C=log|cosh(x)|+C

倒數雙曲函數

  • csch(x)dx=icsc(ix)dx=log|itan(ix2)|+C=log|tanh(x2)|+C
  • sech(x)dx=sec(ix)dx=2artanh(itan(x2i))+C=2arctan(tanh(x2))+C
  • coth(x)dx=icot(ix)dx=log|isin(ix)|+C=log|sinh(x)|+C

反雙曲函數

  • arsinh(x)dx=xarsinh(x)x2+1+C
  • arcosh(x)dx=xarcosh(x)x21+C
  • artanh(x)dx=xartanh(x)+12ln(1x2)+C
  • arcsch(x)dx=xarcsch(x)+|arsinh(x)|+C
  • arsech(x)dx=xarsech(x)+arcsin(x)+C
  • artanh(x)dx=xarcoth(x)+12ln(x21)+C

雜項

  • |f(x)|dx=sgn(f(x))f(x)dx,其中sgn符号函数

定積分

  • [0,1]ni=1ndxi1i=1nxi=ζ(n),其中整數n>1ζ黎曼ζ函數
  • ex2dx=π
  • 01tu1(1t)v1dt=β(u,v)=Γ(u)Γ(v)Γ(u+v),其中ΓΓ函數
  • 0ts1etdt=Γ(s)
  • 02πeucosθdθ=2πI0(u),其中I0第一類修正貝塞爾函數
  • 0sin(x)xdx=π2

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