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Math Help - help with integral

  1. #1
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    help with integral

    hey guys,

    I need to prove that the following integral converges

    int ( sqrt(x) * cos(x^2) dx , x = 0 to infinity )

    Using Wolfram I found that the solution can be expressed in the closed form

    (1/4) * sqrt ( 1 - sqrt(2) ) * Gamma ( 3 / 4 )

    Whilst I don't have to show the exact result, I am completely lost with showing that it converges.

    Any starting points/ hints would be greatly appreciated,

    Thanks,

    Tim
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  2. #2
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    Opalg's Avatar
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    Quote Originally Posted by Mathisfun3 View Post
    I need to prove that the following integral converges

    \int_0^\infty\!\!\! \sqrt{x}\cos(x^2)\,dx
    Here are a few hints. The function is continuous and bounded on any finite interval, so the only trouble can occur "at infinity". It's convenient to change the lower limit of integration away from 0 to say 1, so that we don't have to worry about what happens at 0. Replace the upper limit of integration by X. Then we want to show that \int_1^X\!\!\! \sqrt{x}\cos(x^2)\,dx converges as X\to\infty.

    Make the substitution y=x^2, and the integral becomes \int_1^Y\!\!\! \tfrac12y^{-1/4}\cos y\,dy, where Y=X^2. Now integrate by parts, integrating the factor \cos y and differentiating y^{-1/4}. That will give you a new integral, which you should be able to estimate to see that it converges as Y\to\infty.
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