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Math Help - Another Gaussian integral...

  1. #1
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    Another Gaussian integral...

    How do you solve this?

    <br />
\int^{\infty}_{0} x^2 e^{-x^2}<br />
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by chutsu View Post
    How do you solve this?

    <br />
\int^{\infty}_{0} x^2 e^{-x^2}<br />
    Integration by parts will reduce leave a known definite integral (splitting the integrand (x)(xe^{-x^2}) )

    CB
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  3. #3
    Super Member Deadstar's Avatar
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    What CaptainBlack said but I couldn't resist posting this...

    Set t=x^2, x = \sqrt{t}, dx = \frac{1}{2}t^{-\tfrac{1}{2}}.

    Integral becomes

    \frac{1}{2} \int_0^{\infty} t^{\tfrac{1}{2}} e^{-t} dt = \tfrac{1}{2}\Gamma(\tfrac{3}{2}) = \tfrac{1}{4}\Gamma(\tfrac{1}{2}) = \tfrac{1}{4}\sqrt{\pi}

    Gamma function - Wikipedia, the free encyclopedia
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