01 October, 2009

Integrals with exponent and polynomial: Pn(x)e^{ax}

I've found a very cute way to work with integrals like $$ \int_{}^{}P_n(x)e^{ax}dx $$ Before I always integrated it by parts, for example: $$ \int_{}^{}xe^xdx=xe^x-\int_{}^{}e^xdx=(x-1)e^x $$ In case you need to take by parts only once it's ok, but when Pn(x) is x^2 or higher degree you need more time to integrate by parts several times.
There is another better way, let's take above integral again:
$$ \int_{}^{}xe^xdx=(Ax+C)e^x $$ So now you should take a derivative of the right side and solve it against left side: $$ xe^x=Ae^x+(Ax+C)e^x $$ If you divide by e^x, you see that A=1 and C=-1 so you get the same correct answer as above (but easier!) So in common case: $$ \int_{}^{}P_n(x)e^{ax}dx=Q_n(x)e^{ax} $$ Where $P_n(x)$ is polynomial of degree $n$ and $Q_n(x)$ is polynomial of degree $n$ with unknown coefficients to find using equation: $$ P_n(x)e^{ax}=Q'_n(x)e^{ax} + Q_n(x)ae^{ax} $$

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