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

[Raincloud]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

The substitution gives
$\int^\infty_\infty exp(-(2y^2-2\sqrt{ab}) dx$
and $dy/dx = 1/2(\sqrt{a} + \sqrt{b}/x^2)$
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
$\sqrt{2\pi/a}exp(-ab)$

 

[Ray Vickson]

Homework Statement

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

Homework Equations

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

The Attempt at a Solution

The substitution gives
$\int^\infty_\infty exp(-(2y^2-2\sqrt{ab}) dx$
and $dy/dx = 1/2(\sqrt{a} + \sqrt{b}/x^2)$
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
$\sqrt{2\pi/a}exp(-ab)$

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

RGV

 

[Raincloud]

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

RGV

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