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Gaussian Integral (almost)

  1. Apr 21, 2012 #1
    1. The problem statement, all variables and given/known data

    I'm having difficulty solving the following integral.

    [itex]\int_{-\infty}^{\infty} x^{4}e^{-2\alpha x^{2}} \text{d}x[/itex]

    2. Relevant equations

    [itex]\int_{-\infty}^{\infty} e^{-\alpha x^{2}} \text{d}x = \sqrt{\frac{\pi}{\alpha}}[/itex]

    [itex]\int_{-\infty}^{\infty} x^{2}e^{-\alpha x^{2}} \text{d}x = \frac{\sqrt\pi}{2\alpha^{\frac{3}{2}}}[/itex]

    3. The attempt at a solution

    I solved a very similar integral, however this one was much easier as I could use substitution quite easily. (In this case I let [itex]u = x^{2}[/itex] and [itex]\text{d}u = 2x\text{d}x[/itex]).

    [itex]\int_{-\infty}^{\infty} x^{3}e^{-2\alpha x^{2}} \text{d}x[/itex]

    (It turns out that the integral above is zero.)

    The only difference in this new question is a factor of x, yet I have no idea how to approach it. Ideally I would want to find a way to manipulate this integral to be of a form that I know the solution (e.g the ones I've listed above).

    Just need a point in the right direction. Any help would be greatly appreciated!
  2. jcsd
  3. Apr 22, 2012 #2
    Are you familiar with this? If you integrate what you have by alpha, you'll get the second relevant equation.
  4. Apr 22, 2012 #3
    I haven't been taught that yet. Is that the only way to solve this integral? I've been reading the related articles but can't work out how to apply it to this question.

    Unfortunately it seems my physics course is ahead of my maths course.
  5. Apr 22, 2012 #4


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    Gold Member

    Take the second relevant equation and do integration by parts - integrate the x2 and differentiate the e-2a x^2 to get an x4 e-2ax^2 integral. Then solve for this new integral in terms of things that you know
  6. Apr 22, 2012 #5
    Took your advice and the answer popped out straight away. How did I not think of this?

    Thanks mate :)
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