calculus - Feynman technique of integration for $int^infty_0 expleft(frac{-x^2}{y^2}-y^2right) dx$

I've been learning a technique that Feynman describes in some of his books to integrate. The source can be found here:

http://ocw.mit.edu/courses/mathematics/18-304-undergraduate-seminar-in-discrete-mathematics-spring-2006/projects/integratnfeynman.pdf

The first few examples are integrals in $x$ and $y$ variables, and I can't see a good way to simplify them using differentiation, particularly the example:

$$\int^\infty_0 \exp\left(\frac{-x^2}{y^2}-y^2\right) dx$$