Integrating rational functions of sine and cosine using the rules of Bioche

Abstract

For integrals consisting of rational functions of sine and cosine a set of little known rules known as the Bioche rules are considered. The rules, which consist of testing the differential form of the integral for invariance under one of three simple substitutions $x\mapsto -x,\pi -x$, and $\pi +x$, allow one to decide which of the three trigonometric substitutions of: $t=\sin x,\cos x$, or $\tan x$ would be most appropriate for finding the integral. In seeking to not only bring these simple rules to the attention of a wider audience, formal justification of the rules is provided and two examples illustrating the use of the rules given.

Keywords:

• Elementary integration
• rational trigonometric functions
• trigonometric substitutions
• Bioche rules

2010 Mathematics Subject Classification:

• 97I50

Acknowledgments

Section 5 resulted from a suggestion by one of the anonymous referees for applying the rules of Bioche. The author is grateful for having received this.

Disclosure statement

No potential conflict of interest was reported by the author.