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Math Help - Need help in Integral

  1. #1
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    Question Need help in Integral

    ∫x^(4)e^(-x^(2)/2)dx in the limit [-∞,∞]

    I tried to solve this problem many times but i just couldn't . Can someone help me ?
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  2. #2
    MHF Contributor harish21's Avatar
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    Re: Need help in Integral

    Quote Originally Posted by nirmal019 View Post
    ∫x^(4)e^(-x^(2)/2)dx in the limit [-∞,∞]

    I tried to solve this problem many times but i just couldn't . Can someone help me ?
    write down the even function as

    \int_{-\infty}^{\infty} x^4 e^{-\frac{x^2}{2}} dx

    =2 \int_0^{\infty} x^4 e^{-\frac{x^2}{2}} dx

    let u=x^2 \implies du = 2x dx. also, x=0 \implies u=0; and x=\infty \implies u=\inftyso you have

    \int_0^{\infty} u^{3/2} e^{-\frac{u}{2}} du

    try solving from here...use gamma function
    Last edited by harish21; August 27th 2012 at 05:56 PM.
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  3. #3
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    Re: Need help in Integral

    Thank you so much I have solved the promblem . Actually I never thought of the Gamma function and trying different methods :P
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  4. #4
    Senior Member MaxJasper's Avatar
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    Lightbulb Re: Need help in Integral

    Using Laplace transform:

    \mathcal{L}_x\left[\int_{-\infty }^{\infty } x^4 e^{-\frac{x^2}{2}} \, dx\right](s) = \frac{3 \sqrt{2 \pi }}{s}

    Taking inverse Laplace:

    \mathcal{L}_s^{-1}\left[\frac{3 \sqrt{2 \pi }}{s}\right](x) = 3 \sqrt{2 \pi }
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