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Thread: Stumped on an integral

  1. #1
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    Stumped on an integral

    Working on a Fourier cosine transform and need a solution to the following integral to continue. Very grateful for any responses/ advice. My integration is a little rusty.

    Integral ((Sin 2pix Cos piX) / X)

    thanks in advance,
    -Dave
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  2. #2
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by DMessin View Post
    Working on a Fourier cosine transform and need a solution to the following integral to continue. Very grateful for any responses/ advice. My integration is a little rusty.

    Integral ((Sin 2pix Cos piX) / X)

    thanks in advance,
    -Dave
    What are the bounds of the integral? If it is this

    $\displaystyle \int_0^{\infty}\frac{\sin(2\pi{x})\cos(\pi{x})}{x} dx$

    The answer is $\displaystyle \frac{\pi}{2}$

    To see this rewrite your integral as

    $\displaystyle \begin{aligned}\int_0^{\infty}\frac{\sin(2\pi{x})\ cos(\pi{x})}{x}dx&=\int_0^{\infty}\sin(2\pi{x})\co s(\pi{x})\int_0^{\infty}e^{-yx}dydx\\
    &=\int_0^{\infty}\int_0^{\infty}e^{-yx}\cos(\pi{x})\sin(2\pi{x})dydx
    \end{aligned}$

    And you may switch the integration order because it is over a rectangular region by Fubini's Theorem.
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  3. #3
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    THANK YOU!!!!



    Brilliant, thanks!!!!!!!
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