$\int_0^{\infty}x^n e^{-ax}dx$

Question

How is the following formula proven?

I tried to use the integral $f(x)g'(x) (g(x) = \text{gamma function)}$ but it doesn't work.

This formula is used in Planck function integral and I really need to understand it for my astrophysics class. Any help is appreciated.

Answer 1

Recall that

$$\Gamma(n)=\int_0^\infty x^{n-1}e^{-x}~\mathrm{d}x.$$

Now let $t=ax$, $\mathrm{d}t=a~\mathrm{d}x$ in your integral, which gives you that

$$\int_0^\infty x^ne^{-ax}~\mathrm{d}x=\int_0^\infty \left(\frac{t}{a}\right)^ne^{-t}\frac{1}{a}~\mathrm{d}t=\frac{1}{a^{n+1}}\int_0^\infty t^ne^{-t}~\mathrm{d}t=\frac{\Gamma(n+1)}{a^{n+1}}.$$