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Math Help - Integral proof ..

  1. #1
    Junior Member BayernMunich's Avatar
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    Integral proof ..

    Hello,

    Problem:

    Show that:

    \int_0^1 x^m \, ln^n(x) \, dx = (-1)^n \, \dfrac{n!}{(m+1)^{n+1}}

    where m and n are nonnegative integers..
    I want a start, maybe a substitution?
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  2. #2
    Junior Member BayernMunich's Avatar
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    Am focus on the gamma function..

    I used the substitution u=-ln(x) ..

    The integral will be: \int_0^{\infty} (-1)^n t^n e^{-(m+1)t} \, dt


    I stopped here, any idea?
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  3. #3
    Junior Member BayernMunich's Avatar
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    Thanks, I solved it =)

    The last step is to substitute z=(m+1)t and recalling that \Gamma(n+1)=n! ..
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