Feynman’s Favorite Math Trick
Differentiating under the integral sign
Feynman is no doubt one of the tyrants in the field of Quantum mechanics. His peculiar curiosity in any intellectual field was unfathomable. In Mathematics, his favorite trick was to use differentiation under the integral sign. Derivatives are structured and easier to analyze than integration, thus Feynman percolated the idea, ‘Why not use differentiation within the integration to make integrals easier to solve?’
A slight change in the structure of a function in integrals makes it far too difficult to solve it. for eg.
This is a commonly structured integral whose solution is known to all, which is arctanx + c. But if we ever so slightly change the power of x, say
This should be easy right? as it is in some ways of the same format? Wrong. The analysis becomes menacingly harder than the previous problem and so does the solution. The solution would be
Feynman burrowed the simpler ideas from differentiation and used them to solve the complex integral. It was his favorite trick in the field of mathematics. A sample solution to an integral problem using the Feynman technique is given as:
We could solve this integral by the traditional method ‘Integration by parts’ and following the ILATE rule. But this would be time-consuming and might not work in many cases. Instead applying Feynman’s trick given as:
We all know,
What we do is, we add a random variable ‘a’ to the ‘x’
So the integral becomes,
Now, we are going to differentiate the function with respect to ‘a’ under the integral sign,
Recognizing the pattern of the solution, in general, we could write,
As ‘a’ is a random variable assigned by us, we could take any value for it. Let’s assume it's ‘1’.
So the integral becomes,
Which is the solution to the integral problem using the Feynman technique.
Other solutions to complex integral problems made easy by Feynman’s trick are available online.
Contributed by Rishab Karki and curated by the author.
Thank you so much for reading. If you liked this story don’t forget to press that clap icon. If you like my works and want to support me then you can become a medium member by using this link or buy me a coffee ☕️. Keep following for more such stories.
