Integrals of Functions of Roots of \(x\)

Resumen

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.