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Dealing w/slight modification of Gaussian integral?

  1. Feb 24, 2016 #1
    Hey, folks. I'm doing a problem wherein I have to evaluate a slight variation of the Gaussian integral for the first time, but I'm not totally sure how to go about it; this is part of an integration by parts problem where the dv is similar to a gaussian integral:

    [tex] \int_{-\infty}^{\infty}x^2e^{-\alpha x^2} dx, \ \text{where} \ \alpha > 0 [/tex]. My question is, how should I deal with the alpha in the exponent? I know the Gaussian integral equals
    [tex] \sqrt{\pi}[/tex]

    But how do I deal with that alpha?

    I thought I found my answer in a handout, but when I applied it, I ended up still being wrong; I attempted to integrate by parts and I got zero, the two terms cancelling each other out; what gives?
    Last edited: Feb 24, 2016
  2. jcsd
  3. Feb 25, 2016 #2


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    Science Advisor
    Homework Helper

    Try a substitution to get back to the basic form: ##y=\sqrt{\alpha} x##
    Then integration by parts.
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