I'm trying to calculate the integral (which is a function of a):

$\displaystyle \int_0^\infty(x^2+a)^{-n}dx$, where $\displaystyle n$ is a positive integer and $\displaystyle a>1$

The aim of the problem is to do it by differentiating through the integral sign, but i've done that with respect to a, and I've just got

$\displaystyle \int_0^\infty-n(x^2+a)^{-n-1}dx$ which is no easier to solve.

What's the best way to go about this?