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Math Help - integral with log and sinh

  1. #1
    Eater of Worlds
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    integral with log and sinh

    Give this one a go if you're so inclined. A clever sub will do it.

    \int_{0}^{\infty}\int_{y}^{\infty}\frac{(x-y)^{2}log(\frac{x+y}{x-y})}{xysinh(x+y)}dxdy
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  2. #2
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by galactus View Post
    Give this one a go if you're so inclined. A clever sub will do it.

    \int_{0}^{\infty}\int_{y}^{\infty}\frac{(x-y)^{2}log(\frac{x+y}{x-y})}{xysinh(x+y)}dxdy
    I thought I had seen this before...and I searched, and was correct!

    I remember working on this to no avail...plus, it turned out to be beyond the scope of what I knew anyway... >_>

    --Chris
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  3. #3
    Math Engineering Student
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    Here.
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  4. #4
    Eater of Worlds
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    Here's another way.

    An alternative sub is to let x=\frac{u+t}{2}, \;\ y=\frac{u-t}{2}

    Then, we get:

    I=2\int_{0}^{\infty}\frac{1}{sinh(u)}\int_{0}^{u}\  frac{t^{2}log(\frac{u}{t})}{u^{2}-t^{2}}dtdu

    Then, in the inner integral, let t=uw to get the product of two known integrals.

    I=2\int_{0}^{\infty}\frac{udu}{sinh(u)}\int_{0}^{1  }\left(1-\frac{1}{1-w^{2}}\right)log(w)dw

    And you should get:

    \frac{{\pi}^{2}({\pi}^{2}-8)}{16}
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