## 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: $$P_n(x)e^{ax}=Q'_n(x)e^{ax} + Q_n(x)ae^{ax}$$