Feynman's Trick a.k.a. Differentiation under the Integral Sign & Leibniz Integral Rule
Among a few other integral tricks and techniques, Feynman's trick was a strong reason that made me love evaluating integrals, and although the technique itself goes back to Leibniz being commonly known as the Leibniz integral rule, it was Richard Feynman who popularized it, which is why it is also referred to as Feynman's trick. Here's an excerpt from his book, Surely You're Joking, Mr. Feynman:
"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.
One day he told me to stay after class. "Feynman," he said, "you talk too much and you make too much noise. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again."
So every physics class, I paid no attention to what was going on with Pascal's Law, or whatever they were doing. I was up in the back with this book: Advanced Calculus, by Woods. Bader knew I had studied Calculus for the Practical Man a little bit, so he gave me the real works -- it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions -- all kinds of wonderful stuff that I didn't know anything about.
That 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, 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."
For me, employing this trick felt like I was using cheat codes to deal with integrals. At the same time, it enabled a lot of creativity and wishful thinking, which transformed integrals into puzzles. Unfortunately, this also means that there is no clear path on how and when to use this technique. In addition, what Feynman wrote still applies today since the method isn't taught much, if at all, in universities. Therefore, the trick can seem obscure and difficult to grasp for newcomers.
In the following section, we will embark on a journey to develop some rules of thumb to have at our disposal when using Feynman's trick. These are merely some heuristics that I tend to use, so deviating from them can be perfectly acceptable. However, I hope that they can provide a path to follow when nothing obvious or intuitive occurs when someone tries to use this trick, or even better, so that they can serve as motivation for someone to start using the method.
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