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Math Help - Tricky Integral

  1. #1
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    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?
    Last edited by StaryNight; March 8th 2013 at 11:27 AM.
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  2. #2
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    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!
    Last edited by StaryNight; March 8th 2013 at 01:25 PM.
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  3. #3
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    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)!
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