(adsbygoogle = window.adsbygoogle || []).push({}); 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]

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# Homework Help: Integrate exp(-1/2( ax^2 - b/x^2)

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