computation using gaussian integral

Question

Using that the Gaussian integral $\int_0^{\infty}e^{-x^2}\;dx$ is equal to $\sqrt{\pi}/2$, compute the following integral $$ \int_0^{\infty}e^{-\left(x^2+\frac{a^2}{x^2}\right)}\;dx$$ where $a>0$ is a parameter.

Answer 1

Let $u:=x-a/x$ so, by Glasser's master theorem, your integral is$$\frac12e^{-2a}\int_{\Bbb R}e^{-u^2}dx=\frac12e^{-2a}\int_{\Bbb R}e^{-u^2}du=\frac{\sqrt{\pi}}{2}e^{-2a}.$$