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Math Help - Integral of x cotan (x) dx from 0 to pi/2

  1. #1
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    Integral of x cotan (x) dx from 0 to pi/2

    Integral of x cotan (x) dx from 0 to pi/2

    Please solve using all steps. thanks!!
    Last edited by mr fantastic; May 28th 2010 at 03:13 PM. Reason: Re-titled.
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  2. #2
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    Did you try doing this 'under the Integral'?. This one is often done that way.

    xcot(x)=\frac{x}{tan(x)}

    I(a)=\int_{0}^{\frac{\pi}{2}}\frac{tan^{-1}(a\cdot tan(x))}{tan(x)}dx

    I'(a)=\frac{d}{dx}\int_{0}^{\frac{\pi}{2}}\frac{ta  n^{-1}(a\cdot tan(x))}{tan(x)}dx

    =\int_{0}^{\frac{\pi}{2}}\frac{\partial}{{\partial  }b}\left[\frac{tan^{-1}(a\cdot tan(x))}{tan(x)}\right]dx

    \int_{0}^{\frac{\pi}{2}}\frac{1}{(a\cdot tan(x))^{2}+1}dx

    =\frac{\pi}{2(a+1)}

    Now, integrate w.r.t a. Then, let a=1 to finish line.

    This is a cool method. Learn about 'differentiating under the integral sign'.
    You can probably find it somewhere.
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