Conclusion
Overall, Richard Feynman introduces such a unique and simple technique to mathematics and has made calculations and computations of integrals uncomplicated. The idea of differentiating under the integral sign, if used properly, can be used to solve any difficult integral. On further studying higher-level math classes, like Real Analysis and Complex Analysis, Mathematicians and Physicists can solve their required problem with Feynman's Integral technique by tweaking their functions or equations problems in Banach space a lot simpler. Physicists dealing with quantum mechanics use techniques like these to deal with path integrals. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional integral, over an infinity of quantum-mechanically possible trajectories to compute a quantum amplitude. The technique helps them generalize the action principle of classical mechanics in physics when dealing with quantum mechanics.
As we have seen over these past few slides, there are many ways to apply Feynman's integral technique, both mathematically and towards other subjects. Although this integration technique focuses mainly on manipulating functions and differentiating to simplify a problem, there are many different ways to think about this substitution, and it can fit numerous situations. The first time we saw this use of the theory behind Feynman's rule, rather than the rule itself, was in the "Various Uses" section, where we approached this technique from a unique angle. Just like any other mathematical concept we learned, this integration rule is best used after fully understanding the reasoning behind it, instead of simply copying down the procedure and using it mindlessly. The assistance that it can provide is based on your ability to apply it in different situations, allowing it to extend towards other topics inside of mathematics and other fields such as physics.
