Evaluation of specific integrals by differentiation – part 2

Norbert Kecskés

doi:10.15414/meraa.2021.07.01.10-15

Abstract

One of the most important computational techniques in higher mathematics is differentiation and its counterpart, integration (anti-differentiation). While differentiation is a routine and relatively simple procedure, integration, in general, is a much more involving task. Close (inverse) relationship between differentiation and anti-differentiation (evaluation of indefinite integrals) in some cases reveals the possibility to derive the form of the antiderivative and evaluate this antiderivative by differentiation and subsequent comparison of coefficients. This paper is a sequel to [4] and deals with some other types of elementary functions whose integrals can be evaluated by differentiation.

Highlights

Integration is a widely used technique in calculus
In [4], motivated by [3], we investigated a family of simple elementary functions whose antiderivatives can be found by differentiation and comparison of coefficients
Since the square root of a linear term is transformed by differentiation into its reciprocal, we will consider the function Qn x ax b and its derivative

Summary

Introduction

Integration is a widely used technique in calculus. Various standard methods of evaluation of elementary indefinite integrals, e. g. tabular, substitution, integration by parts or partial fraction decomposition can be found in [1], [2], [5]. In [4], motivated by [3], we investigated a family of simple elementary functions whose antiderivatives can be found by differentiation and comparison of coefficients. We discussed the antiderivatives of the following functions: Pn x ea x , Pn x sin ax , Pn x cos ax , ea x sin bx , ea x cos bx, where Pn x denotes a polynomial of n–th degree.

Results

Conclusion