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Math Help - Problem #1 - Integration

  1. #1
    Forum Admin topsquark's Avatar
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    Problem #1 - Integration

    Solve the integral:
    \int _0 ^{\pi} \frac{x~sin(x)}{1 + cos^2(x)} dx

    (Source: http://www.math.utah.edu/)
    Last edited by topsquark; April 16th 2014 at 08:57 AM.
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  2. #2
    Forum Admin topsquark's Avatar
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    re: Problem #1 - Integration

    Well, no one answered. It's a simple enough problem to do by integration by parts but I like the solution I found as it doesn't have to do with parts.

    \int_0^{\pi} \frac{x~sin(x)}{1 + cos^2 (x)} dx

    Let x = \pi - z giving dx = - dz.

    Then
    \int_0^{\pi} \frac{x~sin(x)}{1 + cos^2 (x)} dx = - \int _{\pi}^0 \frac{( \pi - z ) sin( \pi - z )}{1 + cos^2( \pi - z )} dz

    After some work with the addition of angles formulas:

    \int_0^{\pi} \frac{x~sin(x)}{1 + cos^2 (x)} dx = \int_0^{\pi} \frac{ \pi sin( z )}{1 + cos^2 (z) } dz - \int_0^{ \pi } \frac{z~sin(z)}{1 + cos^2(z)} dz

    x and z are "dummy" variables so change the z on the RHS to x and move the last term on the right to the LHS:
    2 \int_0^{\pi} \frac{x~sin(x)}{1 + cos^2 (x)} dx = \int_0^{\pi} \frac{ \pi sin( z )}{1 + cos^2 (z) } dz

    The integral on the RHS is elementary and gives:
    \int_0^{\pi} \frac{x~sin(x)}{1 + cos^2 (x)} dx = \frac{ \pi }{2} \cdot \left ( \frac{ \pi }{2} \right ) = \left ( \frac{ \pi }{2} \right ) ^2

    -Dan
    Thanks from HallsofIvy and Jonroberts74
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