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Math Help - n-fold integral

  1. #1
    Per
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    n-fold integral

    Can anyone give me any suggestion of how to solve this integral? Or better - has anyone seen any solution to this kind of integral?


    <br />
\int_0^1\int_0^1\cdots \int_0^1\frac{1}{\left( c+x_{1}+x_{2}+\cdots +x_{n}\right) }dx_{1}\cdots dx_{n}<br />

    I would be grateful for any help!
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  2. #2
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    Quote Originally Posted by Per View Post
    Can anyone give me any suggestion of how to solve this integral? Or better - has anyone seen any solution to this kind of integral?


    <br />
\int_0^1\int_0^1\cdots \int_0^1\frac{1}{\left( c+x_{1}+x_{2}+\cdots +x_{n}\right) }dx_{1}\cdots dx_{n}<br />

    I would be grateful for any help!

    Assuming c+x_1+\ldots+x_n>0\,,\,\,\forall \,0\leq x_i\leq 1 :

    Beginning with the first one: \int^1_0\frac{1}{c+x_1+\ldots +x_n}\,dx_n=[\ln(c+x_1+\ldots +x_n)]^1_0= \ln(c+x_1+\ldots +x_{n-1}+1)-\ln(c+x_1+\ldots +x_{n-1}) ... I purposedly

    didn't write this as the logarithm of a quotient since for the next step is easier to integrate this way.

    Next step: \int^1_0\left(\ln(c+x_1+\ldots +x_{n-1}+1)-\ln(c+x_1+\ldots +x_{n-1})\right)\,dx_{n-1}= \left[(c+x_1+\ldots +x_{n-1}+1)(\ln(c+x_1+\ldots +x_{n-1}+1)-1)\right]^1_0

    -\left[(c+x_1+\ldots +x_{n-1})(\ln(c+x_1+\ldots +x_{n-1})-1)\right]^1_0= ...etc.

    The above doesn't look too hard but it looks pretty messy and ugly...I don't know if there's a nice expression for it all.

    Tonio
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