Method of differentiating under the integral sign

''I could never resist a definite integral'' (G. H. Hardy - Coxeter, 1968) ''One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high-school physics teacher Mr. Bader had given me.... The book also showed how to differentiate parameters under the integral sign - It's a certain operation. It turns out that's not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was that, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me.'' (Richard P. Feynman, 1985)