Integrals of functions containing parameters

TL;DR

This paper proposes a correction to the standard practice of computing indefinite integrals, introducing the 'Kahanian constant of integration,' which has significant implications for education and computer algebra systems.

Contribution

It introduces the 'Kahanian constant of integration' as a correction to current integral computation methods, impacting teaching and software implementations.

Findings

Current practices propagate minor errors in integral calculations.

The correction improves the accuracy of computer algebra systems.

Educational practices may need revision based on this correction.

Abstract

This paper offers what seems at first to be a minor technical correction to the current practice of computing indefinite integrals, and introduces the idea of a "Kahanian constant of integration". However, the total impact of this minor correction is potentially large because the current practice is taught early at the university level and to very many students---most of whom do not go on to become mathematics majors. Moreover, computer algebra systems have become widespread, including good free ones, some of which are available for smartphones. Most current computer algebra systems apply current textbook rules and amplify the effects of fundamental "minor" errors such as the error in continuity that we address in this article. So in practice, the correction we present is important.