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Integral ((4x^2)-1)e^(-2x^2)

  1. Feb 27, 2009 #1
    I was wondering if anyone could point me in the right direction with this integral.

    ((4x^2)-1)e^(-2x^2),

    I have tried substitution with trig functions, hyperbolic functions, seperating the first part into partial fractions and numerous other methods to no avail. Does anyone who knows have any hints?

    Cheers
     
  2. jcsd
  3. Feb 27, 2009 #2
    Try solving this with the http://en.wikipedia.org/wiki/Error_function" [Broken]. This will give a solution in terms of erf(x), a solution in terms of more elementary functions is probably not possible.
    I also suspect that the definite integral over the real line is 0.
     
    Last edited by a moderator: May 4, 2017
  4. Feb 27, 2009 #3
    It is OK I have done it now, I split it in to two parts 4x^2(e-2x^2) and -e(-2x^2) and integrated the first part by parts using u'=4xe(-2x^2) and v=x, and then the integral part of the solution cancelled with the second part ie the sol'n was ;

    [-xe(-2x^2)]-int(-e(-2x^2)-int(-e(-2x^2)),

    leaving [-xe(-2x^2)] with appropriate limits,

    Thanks!
     
  5. Feb 27, 2009 #4

    HallsofIvy

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    No, the last two terms do NOT cancel, they add. This is
    [tex]-xe^{-2x^2}- 2\int e^{-2x^2} dx[/tex].

     
  6. Feb 27, 2009 #5
    I think wrote it incorrectly, it would be;

    (-xe^(-2x^2)) -int(-e^(-2x^2)) -int(e^(-2x^2)),

    which then cancels.
     
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