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

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 , and , allow one to decide which o...

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Bibliographic Details
Published in:International journal of mathematical education in science and technology 2022-06, Vol.53 (6), p.1688-1700
Main Author: Stewart, Seán M.
Format: Article
Language:English
Subjects:
Bioche rules
Calculus
Elementary integration
Equations (Mathematics)
Mathematical Concepts
Mathematics education
Mathematics Instruction
Problem Solving
Rational functions
rational trigonometric functions
Trigonometric functions
trigonometric substitutions
Trigonometry
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Summary: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 , and , allow one to decide which of the three trigonometric substitutions of: , or 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.
ISSN:0020-739X
1464-5211
DOI:10.1080/0020739X.2021.1912841