The coolest way to evaluate the Fresnel integrals (Feynman's technique)

Although this is awesome I still wouldn't call it the most elegant; that would be using complex analysis and an analogy with laplace transforms as in the following video:

A beautiful approach to the Fresnel i...

And in case you don't want to start with complex analysis, a laplace transform applied to either Fresnel integral can work too:

Asking Laplace to help us solve the F...