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Integrate exp(-1/2( ax^2 - b/x^2)

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

    Integrate:
    [itex] I(a,b) =
    \int^\infty_\infty exp(-1/2(ax^2+b/x^2)) dx [/itex]
    given
    [itex]\int^\infty_\infty exp(1x^2/2) dx = \sqrt{2\pi} [/itex]


    2. Relevant equations

    The suggested substitution is [itex] y = (\sqrt{a}x - \sqrt{b}/x)/2 [/itex]

    3. The attempt at a solution

    The substitution gives
    [itex]\int^\infty_\infty exp(-(2y^2-2\sqrt{ab}) dx [/itex]
    and [itex] dy/dx = 1/2(\sqrt{a} + \sqrt{b}/x^2) [/itex]
    but I can't seem to rearrange the dy/dx to do anything helpful. I've tried integrating by parts before plugging in the substitution, but it didn't seem to help.

    FWIW, i've been told the numerical answer is
    [itex] \sqrt{2\pi/a}exp(-ab)[/itex]
     
  2. jcsd
  3. Jan 21, 2012 #2

    Ray Vickson

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    Maple gets [tex]\sqrt{\frac{2\pi}{a}} \;e^{-\sqrt{ab}},[/tex] whiich is not what you wrote.

    RGV
     
  4. Jan 21, 2012 #3


    Yes, you're right- looking at my notes I have lost a [tex]\sqrt[/tex] in there.
     
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