Hypergeometric form of Fundamental theorem of calculus

TL;DR

This paper presents a simple, hypergeometric-based method for computing antiderivatives of a broad class of functions, making integration more accessible and potentially simplifying complex integral calculations.

Contribution

It introduces a new, easy-to-teach hypergeometric approach to integration that can handle challenging integrals traditionally solved by heuristic methods.

Findings

Method is straightforward and suitable for undergraduate teaching.

Can prove complex integrals typically approached heuristically.

Highlights the natural connection between antiderivatives and hypergeometric functions.

Abstract

We introduce a natural method of computing antiderivatives of a large class of functions which stems from the observation that the series expansion of an antiderivative differs from the series expansion of the corresponding integrand by just two Pochhammer symbols. All antiderivatives are thus, in a sense, "hypergeometric". And hypergeometric functions are therefore the most natural functions to integrate. This paper would like to make two points: First, the method presented is easy. So much so that it can be taught in undergraduate university level. And second: It may be used to prove some of the more challenging examples computed only by heuristic processes like Method of brackets.