Originally Posted by

**Amanda1990** Here's the question:

By integrating through the integral sign, evaluate $\displaystyle \int_{0}^\infty (x^2 + a)^{-n} dx$ where n is a positive integer and a > 1.

So the standard idea is to let F(a) = $\displaystyle \int_{0}^\infty (x^2 + a)^{-n} dx$ then use (after checking various conditions hold) F'(a) = $\displaystyle \int_{0}^\infty (-n)(x^2 + a)^{-n-1} dx$ by just differentiating through the integral sign, evaluating the integral and solving the resulting differential equation for F(a).

But here we cannot easily work out the integral, so what can we do?