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I have seen Feynman's integral trick coming into use in many questions, but I don't really see a common way to recognize the format for it. Of course when a question comes with an explicit parameter, it's easy to recognize the pattern and follow the steps known to us in the form of Feynman's Trick. So, is there any standard way to quickly recognize the same being in action for any asked question, i.e., how to observe that the question intends us to make use of Feynman's method?

Arsenic
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  • I did try to find a relevant thread for the same, but I couldn't come across anything that would give the clarity needed. But, in case I missed a few threads- I will also appreciate if I could be linked to those threads. – Arsenic Aug 01 '22 at 02:19
  • For 90% of the encountered cases it's useless. It is usable only in some very few niche questions – tryst with freedom Aug 01 '22 at 02:21
  • Yeah, this helps a lot! Are there more threads like these and, if yes, could you link me to them as well? – Arsenic Aug 01 '22 at 02:28
  • I liked this medium article https://www.cantorsparadise.com/richard-feynmans-integral-trick-e7afae85e25c – tryst with freedom Aug 01 '22 at 02:32
  • Thanks, I have gone through this and it's a good read! – Arsenic Aug 01 '22 at 02:34
  • Here's a stupid observation: Feynman's trick is a form of wishful thinking. "If only there was an $e^x$ here," or "if only that constant was something else," then maybe it works. You change the integral to be a family of integrals where one point is the thing you wish it was, and then see if that family is easier to study. – A. Thomas Yerger Aug 01 '22 at 02:38
  • When to differentiate under the integral sign? – Quanto Aug 01 '22 at 02:44
  • Feynman himself said (I think in Surely You're Joking Mr. Feynman) that the other students only brought him an integral if they had all failed to solve it. And that meant that the other students had already tried all the usual tricks. So it was more likely that the "differentiating under the integral" trick would work on those problems, because all the usual tricks had failed. – littleO Aug 01 '22 at 04:17

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Unfortunately, there is no easy way to tell if Feynman's Trick is helpful. You just have to be creative.

One trick I found out is that if you see an integral of the form $\int_a^b \frac{f(h(x))}{h(x)g(x)}$, you can let $I(a) = \int_a^b \frac{f(ah(x))}{h(x)g(x)}$ then take the partial derivative with respect to $a$ to get $I'(a) = \int_a^b \frac{f'(ah(x))}{g(x)}$, which could be easier to work with depending on what $f$ and $g$ are.

Sometimes whenever I try to solve a difficult integral, I choose a parameter inside that integral, go on Symbolab, take the partial derivative with respect to that parameter, and rinse and repeat for different possible parameters to see if I can get an easier function to integrate.

Accelerator
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