I'm trying to show that

$\displaystyle \int_{0}^{\infty}ate^{-at}(1-e^{-at})^{n-2}(1-ne^{-at}) dt = {1 \over {an}} $ for $\displaystyle n \ge 2$

I've tried using induction over n, but I'm not sure this is the best way to go. Is there some substitution I'm missing?