Learning Feynman's Trick for Integrals • Hacker News Board

Re: Learning Feynman's Trick for Integrals

#41

Post by jjtheblunt » Sun, Nov 30, 2025, 6:39 PM UTC

zamadatix wrote: post #21

analog31 wrote: post #14

I don't know if this is exactly the same as what I learned in high school as "integration by substitution." A number of years after I finished school, I was in a new town without a job, and got hired to teach a freshman algebra course at the nearby Big Ten university. About halfway into teaching the class, I was struck by the realization that virtually every problem was solved in the same way, by recognizing the "for…

I think the difference is something like Feynman’s trick simplifies a hard integral by introducing a parameter and differentiating the whole integral, while substitution simplifies an integral by changing variables to undo the chain rule. But it has been so long since I've done integration manually I'm not 100% sure that's an accurate description/the full story. The thing I hated about integration was which approach…

https://en.wikipedia.org/wiki/Risch_algorithm

is super interesting, related to your last sentence.

Re: Learning Feynman's Trick for Integrals

#42

Post by hershkumar » Sun, Nov 30, 2025, 9:58 PM UTC

lordnacho wrote: post #10

zeroonetwothree wrote: post #8

Earlier quoted context omitted.

That’s how most of math works past high school. It requires a lot of practice and intuition.

I don't know about this particular case though, I get the feeling there's a system to it that can be exploited by eg Wolfram. It's just that you're in the dark for a long time before you find the switch.

“Perhaps I could best describe my experience of doing mathematics in terms of entering a dark mansion. One goes into the ﬁrst room, and it’s dark, completely dark. One stumbles around bumping into the furniture, and gradually, you learn where each piece of furniture is, and ﬁnally, after six months or so, you ﬁnd the light switch. You turn it on, and suddenly, it’s all illuminated. You can see exactly where you were.” - Andrew Wiles

Re: Learning Feynman's Trick for Integrals

#43

Post by amai » Mon, Dec 01, 2025, 8:57 AM UTC

zeroonetwothree wrote: post #9

It’s interesting he mentions he doesn’t like contour integration since many integrals can be done either way. Feynman’s trick is equivalent to extending it into a double integral and then switching the order of integration.

Contour integration is basically a double integral in disguise.

Re: Learning Feynman's Trick for Integrals

#44

Post by nextaccountic » Mon, Dec 01, 2025, 11:50 AM UTC

cyberax wrote: post #19

lordnacho wrote: post #10

Earlier quoted context omitted.

I don't know about this particular case though, I get the feeling there's a system to it that can be exploited by eg Wolfram. It's just that you're in the dark for a long time before you find the switch.

Your intuition is right. There is a general algorithm for finding the antiderivatives: https://en.wikipedia.org/wiki/Risch_algorithm Its simplified form can solve pretty much all the undergrad antiderivation problems. I'm a math major, but I consider the time spent learning the tricks for antiderivation to be kinda useless.

Does this makes use of differentiation under the integral sign? (the discussed Feynmann trick)

Re: Learning Feynman's Trick for Integrals

#45

Post by cyberax » Mon, Dec 01, 2025, 10:56 PM UTC

nextaccountic wrote: post #44

cyberax wrote: post #19

Earlier quoted context omitted.

Your intuition is right. There is a general algorithm for finding the antiderivatives: https://en.wikipedia.org/wiki/Risch_algorithm Its simplified form can solve pretty much all the undergrad antiderivation problems. I'm a math major, but I consider the time spent learning the tricks for antiderivation to be kinda useless.

Does this makes use of differentiation under the integral sign? (the discussed Feynmann trick)

It doesn't. But if there is an elementary antiderivative, the Risch algorithm will find it (given the caveats listed in the Wikipedia article). But it might require a lot of substitutions, making its manual application impractical.

Another caveat is that Risch algorithm applies only to antiderivatives, not to the definite integrals. Some definite integrals can be computed without finding the antiderivative, often with the help of Feynman's trick.