The Art Of Problem Solving website is an excellent source of interesting maths problems, although most of them are of an olympiad level (if not higher). Still, it's as good a place as any to find some truly beautiful mathematical problems. Browsing through, I came across this thread, which outlines a method for integration that's too good to pass up. It's employed here to find an integral that is very hard to compute normally (click on the picture to read the text, the quality isn't great but it's readable):



This isn't the most beautiful or elegant thing on the site by any means, but since integrals have always been a soft-spot, I couldn't resist posting this.

Edit: I don't think a partial derivative is required in the little exercise above, a plain ol' ordinary derivative should do just fine.