The Change of Variable Formula Integrals, do they have equal value?

TL;DR

This paper confirms that under finiteness conditions, the two integrals in the change of variable formula for Riemann integrals are equal, extending the result to improper integrals with illustrative examples.

Contribution

It provides a straightforward proof that the two integrals are equal under certain conditions and extends this to improper integrals, clarifying a fundamental aspect of Riemann integration.

Findings

The two change of variable integrals are equal if finite.

The result extends to improper Riemann integrals.

Three examples illustrate the main theorem.

Abstract

Assuming that the two integrals in the Change of Variable Formula for the Riemann integral on the real line are finite, one can rightfully ask if they have equal value. We give a positive answer to this question. The proof is very easy to follow and to keep in mind. The result is extended to improper Riemann integrals. Three examples are given.