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Math Help - Complex integration question!

  1. #1
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    Complex integration question!

    Hi, im really stuck on this question and keep ending up with ridiculous answers, i would be really grateful for any help. thankyou
    Explaining your method compute:

    J=\int_{-\infty }^{\infty }\frac{x^{2}dx}{1+x^{4}}

    For the integral J, simplify your answer until you get an expression involving real numbers only.
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  2. #2
    A Plied Mathematician
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    You can use contour integration. See here for a similar example.
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  3. #3
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     \int_{-\infty}^{\infty} \frac{x^2}{ 1 + x^4 } ~dx


     = \frac{1}{2} \int_{-\infty}^{\infty} \frac{(x^2+1)-(x^2 - 1 ) }{ 1 + x^4 } ~dx

     =  \int_{0}^{\infty} \frac{x^2 + 1 }{ 1 + x^4 } ~dx -  \int_{0}^{\infty} \frac{x^2-1}{ 1 + x^4 } ~dx

    By substituting  x = \frac{1}{t} in the second integral , we find that it is being zero !

    Therefore , what you need to do is the first integral :

     \int_{0}^{\infty} \frac{x^2 + 1 }{ 1 + x^4 } ~dx

     = \int_{0}^{\infty} \frac{1 + 1/x^2}{ x^2  + 1/x^2 } ~dx

     \int_{0}^{\infty} \frac{d(x- 1/x) }{ (x-1/x)^2 + 2  }

     = \int_{-\infty}^{\infty} \frac{du}{u^2 + 2 }

     =  \frac{\pi}{\sqrt{2}}
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