ostensibly discontinuous antiderivative

The real function

x15-3cosx (1)

is continuousMathworldPlanetmathPlanetmath for any x (the denominator is always positive) and therefore it has an antiderivative, defined for all x.  Using the universal trigonometric substitutionPlanetmathPlanetmath


we obtain




This result is not defined in the odd multiples of π, and it seems that the functionMathworldPlanetmath (1) does not have a continuous antiderivative.

However, one can check that the function

xx4+12arctansinx3-cosx+C (2)

is everywhere continuous and has as its derivativePlanetmathPlanetmath the function (1); one has



  • 1 Ernst Lindelöf: Johdatus korkeampaan analyysiin. Fourth edition. Werner Söderström Osakeyhtiö, Porvoo ja Helsinki (1956).
Title ostensibly discontinuous antiderivative
Canonical name OstensiblyDiscontinuousAntiderivative
Date of creation 2013-03-22 18:37:08
Last modified on 2013-03-22 18:37:08
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 7
Author pahio (2872)
Entry type Example
Classification msc 26A36
Related topic CyclometricFunctions