Sunday, May 9, 2010

integration's formula

Composed functions:

  • \int \cos ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( a\sin ax + b\cos ax \right) + C
  • \int \sin ax\, e^{bx}\, dx = \frac{e^{bx}}{a^2+b^2}\left( b\sin ax - a\cos ax \right) + C
  • \int \cos ax\, \cosh bx\, dx = \frac{1}{a^2+b^2}\left( a\sin ax\, \cosh bx+ b\cos ax\, \sinh bx \right) + C
  • \int \sin ax\, \cosh bx\, dx = \frac{1}{a^2+b^2}\left( b\sin ax\, \sinh bx- a\cos ax\, \cosh bx \right) + C


Absolute value functions:
  • \int \left \sin{ax} \right\,dx = {-1 \over a} \left \sin{ax} \right \cot{ax} + C
  • \int \left (ax + b)^n \right\,dx = {(ax + b)^{n+2} \over a(n+1) \left ax + b \right} + C \,\, [\,n\text{ is odd, and } n \neq -1\,]
  • \int \left \cos{ax} \right\,dx = {1 \over a} \left \cos{ax} \right \tan{ax} + C
  • \int \left \tan{ax} \right\,dx = {\tan(ax)[-\ln\left\cos{ax}\right] \over a \left \tan{ax} \right} + C
  • \int \left \csc{ax} \right\,dx = {-\ln \left \csc{ax} + \cot{ax} \right\sin{ax} \over a \left \sin{ax} \right} + C
  • \int \left \sec{ax} \right\,dx = {\ln \left \sec{ax} + \tan{ax} \right \cos{ax} \over a \left \cos{ax} \right} + C
  • \int \left \cot{ax} \right\,dx = {\tan(ax)[\ln\left\sin{ax}\right] \over a \left \tan{ax} \right} + C
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