Vol. 34 (2024)
Articles

Integrals of Functions of Roots of \(x\)

Robert W. Benim
Department of Applied Mathematics, University of Colorado Boulder

Published 2024-01-02

DOI:

https://doi.org/10.71712/kkgf-rv90

Keywords

  • Integration techniques

How to Cite

Benim, R. W. (2024). Integrals of Functions of Roots of \(x\). Scientia Series A: Mathematical Sciences, 34, 23-26. https://doi.org/10.71712/kkgf-rv90

Abstract

This note will show that the only integrals of the form \(\int f(x^\alpha) \, dx\) that can be evaluated via the substitution \(t = x^\alpha\) followed by repeated integration by parts are those where \(\alpha\) is the reciprocal of a positive integer, \(\alpha = \frac{1}{n}\), and \(f\) is an arbitrary function which can be antidifferentiated at least \(n\) times.

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