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Help with Gaussian integration problem please

  1. Sep 1, 2011 #1
    Help with Gaussian integration problem please :)

    1. The problem statement, all variables and given/known data
    Compute the improper integral
    [itex]\int^{\infty}_{-\infty}x^{2}e^{-x^2}dx[/itex]
    given
    [itex]\int^{\infty}_{-\infty}e^{-x^2}dx=\sqrt{\pi}.[/itex]

    2. Relevant equations
    Just the rule for doubly-improper integrals I guess:
    [itex]\int^{\infty}_{-\infty}f(x)dx=\lim_{a\rightarrow-\infty}\lim_{b\rightarrow\infty}\int^{b}_{a}f(x)dx[/itex]

    [itex]erf(x)[/itex] is beyond the scope of this course and thus cannot be utilized in any way.

    3. The attempt at a solution
    I can't see any substitutions that would make things easier, and integration by parts doesn't seem useful (choosing to derive [itex]u=e^{-x^2}[/itex] and integrate [itex]dv=x^{2}dx[/itex] will never simplify or isolate [itex]e^{-x^2}[/itex], and you can't choose to integrate [itex]dv=e^{-x^2}dx[/itex] and derive [itex]u=x^2[/itex] because we are only given the special case of [itex]\int^{\infty}_{-\infty}e^{-x^2}dx[/itex] and not a general antiderivative.
     
  2. jcsd
  3. Sep 1, 2011 #2

    Hootenanny

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    Re: Help with Gaussian integration problem please :)

    I'll give you a little hint.

    [tex]\frac{d}{dx}e^{-x^2} = -2xe^{-x^2}[/tex]

    Hence,

    [tex]\begin{aligned} \int_{-\infty}^\infty x^2 e^{-x^2} \text{d}x & = -\frac{1}{2}\int_{-\infty}^\infty x (-2x e^{-x^2}) \text{d}x \\ & = - \frac{1}{2}\int_{-\infty}^\infty x \frac{d}{dx}e^{-x^2}dx\end{aligned}[/tex]

    Can you now use integration by parts?
     
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