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Math Help - Fresnel Integral

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    Fresnel Integral

    Evaluate \int_0^\infty\cos x^2\,dx
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    MHF Contributor kalagota's Avatar
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    Quote Originally Posted by liyi View Post
    Evaluate \int_0^\infty\cos x^2\,dx
    would this help you?? Fresnel Integrals -- from Wolfram MathWorld
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    Quote Originally Posted by liyi View Post
    Evaluate \int_0^\infty\cos x^2\,dx
    \sqrt{\frac{\pi}{8}} if I remember correctly. Too lazy to do this integral because I done it before.
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    Quote Originally Posted by liyi View Post
    Evaluate \int_0^\infty\cos x^2\,dx
    Substitute u=x^2, the integral becomes to \int_0^\infty {\frac{{\cos u}}{{2\sqrt u }}\,du} .

    The following parameter may be useful: \int_0^\infty {\frac{{e^{ - \alpha u} }}{{\sqrt \alpha }}\,d\alpha } = \frac{{\sqrt \pi }}{{\sqrt u }}.

    To prove this, we can use a substitution and the remarkable fact \int_0^\infty {e^{ - x^2 } \,dx} = \frac{{\sqrt \pi }}{2}.

    Now plug the parameter into the integral, so we'll have a double integral which can be computed without problems.
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