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How to find $$\int_0^\infty e^{-ax^2-\frac{b}{x^2}}dx$$ using gaussian integral?

I tried complete the square: $$-ax^2-\frac{b}{x^2}=-\left(\sqrt{a}x+\frac{\sqrt{b}}{x}\right)^2+2\sqrt{ab}$$, but what next?

I tried integration by parts $u=e^{-ax^2-\frac{b}{x^2}}, dv=dx$, but what next?

eMathHelp
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  • Did you try something? – anumosh May 04 '15 at 10:08
  • Please, see update. – eMathHelp May 04 '15 at 10:16
  • Completing the square was the right thing. But integration by parts won't help. Use the fact $\int _0 ^\infty e^{-x^2} dx = \sqrt \pi /2$ or try to prove it. Here you can find similar question http://math.stackexchange.com/questions/1180168/proof-of-certain-gaussian-integral-form – anumosh May 04 '15 at 10:25
  • I can use the fact you've written. I completed the square, but what to do with $\frac{\sqrt{b}}{x}$? – eMathHelp May 04 '15 at 10:31
  • Found exact same question: http://math.stackexchange.com/questions/496088/how-to-evaluate-int-0-infty-exp-ax2-frac-bx2-dx-for-a-b0 – eMathHelp May 04 '15 at 10:50

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