# Rewriting an integral involving an exponential

1. Aug 23, 2006

### Repetit

Can someone help me show the following:

$$\int_0^{\infty}r^k e^{-a r} dr=\frac{k!}{a^{k+1}}$$

I tried to use the polynomial expansion of e^x:

$$\sum_{n=0,1...} \frac{x^n}{n!}$$

...but I get stuck pretty fast. Can someone give me a few hints?

Thanks!

2. Aug 23, 2006

### matt grime

Just do integration by parts and apply one more well known technique of proof (it might help if you let that integral be I(k), 'cos then what is I(k-1), and how does that link up with the hint to do things by parts?)

3. Aug 23, 2006

### Repetit

I got it now!! I wasn't really as difficult as it first seemed. Thanks alot! :-)