Application of Feynman's Technique
Even though the main concept of this technique has been covered in the What is Feynman's Technique page, there are many extensions of this technique and its uses. One of these applications is brought up by "Mu Prime Math" in his video about Feynman's technique.
The general idea of the method is to set an integral equal to the function of an arbitrary variable in order to differentiate with respect to that variable and receive an integral that is easier to integrate. However, instead of using Feynman’s to set F(t) equal to the integral in the problem and differentiating it to get an easier integral, we can set F(t) equal to an easier integral and differentiate it to get the integral in the problem.
For example, let's take the integral in the video:
While there are other methods of solving this integral, it is also possible to use Feynman's technique to get the answer. However, if we just try and use Feynman's technique normally (by setting a function F(a) equal to $ \int_{0}^{1}(\ln{ax}) dx $) we get a relationship that isn't particularly easy to define. As was explained, Feynman's technique circles back to itself, where we need to be able to find at least one value of F(a), which doesn't seem easy given its current definition.
