Krdytkyu

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:

$$int^infty_0 expleft(frac{-x^2}{y^2}-y^2right) dx$$

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 $yto |x|/y$ so that $dyto -|x|dy/y^2$. 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.

I looked through the paper and see that the "Feynman Technique" is really just a clever application of Leibniz's rule for taking derivatives under integrals. As is pointed out in the comments, the interesting part of this is evaluating $$int_0^infty e^{-x^2} dx$$ using the technique. The paper is a bit misleading for this one since all the other examples added a function of $b$ into the integrand, but that doesn't seem to be the right way to do this one. Instead consider:
$$I(b) = left(int_0^b e^{-x^2} dxright)^2$$
Ultimately you want to evaluate that at $b = infty$ and take its square root.

The full derivation of the result can be found here. That paper discusses this in terms of the Leibniz rule and gives several other interesting derivations of different integrals.