# Tricky Integral

• Mar 8th 2013, 03:09 AM
StaryNight
Tricky Integral
I'm trying to show that

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

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?
• Mar 8th 2013, 12:21 PM
StaryNight
Re: Tricky Integral
I'd be equally glad to know if this is likely to be an unreasonably difficult integral. I obtained the above result using Mathematica and ideally wanted a proof, but it's not essential. Interestingly, the pro online version of Wolfram Alpha runs out of time before being able to evaluate it!
• Mar 8th 2013, 02:11 PM
StaryNight
Re: Tricky Integral
Ok I've figured this out, the strategy is simply to do integration by parts to 'kill' the t-term, after which you are left with a much simpler integration which allows the substitution u=exp(-at)!