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Math Help - [Help] integrate a nasty function

  1. #1
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    [Help] integrate a nasty function

    Dear all,

    Could anyone help me integrate the expression below? Its form is a bit nasty, containing both exponential and power functions.

    F(X) = X^n/[1+exp(X)], n is a positive real number, does not have to be an integer.

    Really urgent to get answer. Thanks!



    Sam
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  2. #2
    Forum Admin topsquark's Avatar
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    Quote Originally Posted by yjwang05 View Post
    Dear all,

    Could anyone help me integrate the expression below? Its form is a bit nasty, containing both exponential and power functions.

    F(X) = X^n/[1+exp(X)], n is a positive real number, does not have to be an integer.

    Really urgent to get answer. Thanks!



    Sam
    There is no solution to this in terms of elementary functions. I'd suggest a power series expansion, then integrate term by term.

    -Dan
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  3. #3
    GAMMA Mathematics
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    Quote Originally Posted by yjwang05 View Post
    Dear all,

    Could anyone help me integrate the expression below? Its form is a bit nasty, containing both exponential and power functions.

    F(X) = X^n/[1+exp(X)], n is a positive real number, does not have to be an integer.

    Really urgent to get answer. Thanks!



    Sam
    F(x)=\frac{x^n}{1+e^x} may not have an elementary solution (doesn't look so outright, so I am trusting topsquark on this one), but

    F(x)=x^n(1+e^x) does have a solution when integrated
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  4. #4
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    If the limits are 0 and \infty then the integral can be expressed in terms of \zeta(n) by writing \frac{1}{1 + e^{x}} as e^{-x} (1 + e^{-x})^{-1} and using the binomial expansion.

    If the limits are t and -t, then the integral will clearly be zero if n is odd; if n is even, it can be found by making the substitution x = -y.
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