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Math Help - Complex Integration

  1. #1
    Junior Member enjam's Avatar
    Joined
    Jul 2009
    Posts
    26

    Complex Integration

    Hey guys, I was wondering if any of you could show me how I would go about using complex integration to show that:

    \int_{0}^{\infty}x^2/(1+x^4)dx = \pi /(2 \sqrt2)

    Thanks.
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  2. #2
    Super Member
    Joined
    Aug 2008
    Posts
    903
    Consider the contour integral over the upper half-disc as the radius extends to infinity:

    \mathop\oint\limits_{C} \frac{z^2}{z^4+1}dz

    By the Residue Theorem, that's equal to:

    \mathop\oint\limits_{C} \frac{z^2}{z^4+1}dz=2\pi i \left(\mathop \text{Res}_{z=e^{\pi i/4}} \frac{z^2}{z^4+1}+\mathop\text{Res}_{z=e^{3\pi i/4}} \frac{z^2}{z^4+1}\right)=\frac{\pi}{\sqrt{2}}

    But the integral is even so:

    \int_0^{\infty}\frac{x^2}{x^4+1}=\frac{1}{2}\frac{  \pi}{\sqrt{2}}

    But I've skipped steps that are important to fully understand the principle.
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