Differentiation under the Integral Sign:In Search of Applications for Feynman's Trick

Abstract

The method of differentiation under the integral sign, popularized by Richard Feynman, is a powerful technique for evaluating complex definite integrals that are resistant to conventional methods. This work demonstrates the applicability and elegance of this method through a fundamental case study in statistics and physics: proving that the probability density functionof the normal distribution is correctly normalized. A direct comparison with the classical method of integration using polar coordinates is presented, highlighting the conceptual and operational simplification offered by Feynman's trick. The analysis concludes that this technique is not only an effective tool for solving specific problems but also fosters a deeper understanding of the interaction between integral and differential calculus.