TITLE: Feynman technique of integration for $\int^\infty_0 \exp\left(\frac{-x^2}{y^2}-y^2\right) dx$
QUESTION [8 upvotes]: 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:
$$\int^\infty_0 \exp\left(\frac{-x^2}{y^2}-y^2\right) dx$$

REPLY [5 votes]: Suppose the integral were $I=\int_0^{\infty} e^{-y^2-\frac{x^2}{y^2}}dy$. Then we note that $y^2+\frac{x^2}{y^2}=\left(y-\frac{|x|}{y}\right)^2+2|x|$.  
Thus, we have
$$I=e^{-2|x|}\int_0^{\infty}e^{-\left(y-\frac{|x|}{y}\right)^2}dy \tag 1$$
Now, substitute $y\to |x|/y$ so that $dy\to -|x|dy/y^2$.  Then, 
$$I=e^{-2|x|}\int_0^{\infty}\frac{|x|}{y^2}e^{-\left(y-\frac{|x|}{y}\right)^2}dy \tag 2$$
If we add $(1)$ and (2), we find
$$\begin{align}
I&=\frac12\,e^{-2|x|}\int_0^{\infty}\left(1+\frac{|x|}{y^2}\right)e^{-\left(y-\frac{|x|}{y}\right)^2}dy \\\\
&=\frac12\,e^{-2|x|}\int_{-\infty}^{\infty}e^{-y^2}dy\\\\
&=e^{-2|x|}\frac{\sqrt{\pi}}{2}
\end{align}$$
So, while not quite a "Feynmann" trick, it is an effective way of evaluation.