On Geometric Interpretations of Euler’s Substitutions

Autor:	Jan L. Cieśliński, Maciej Jurgielewicz
Jazyk:	angličtina
Rok vydání:	2023
Předmět:	integral calculus irrational integrals conics rational parameterization fourth Euler’s substitution Mathematics QA1-939
Zdroj:	Symmetry, Vol 15, Iss 10, p 1932 (2023)
Druh dokumentu:	article
ISSN:	15101932 2073-8994
DOI:	10.3390/sym15101932
Popis:	We consider a classical case of integrals containing an irrational integrand in the form of a square root of a quadratic polynomial. It is known that such “irrational integrals” can be expressed in terms of elementary functions by one of three of Euler’s substitutions. It is less well known that the Euler substitutions have a geometric interpretation. In the framework of this interpretation, one can see that the number 3 is not the most suitable. We show that it is natural to introduce a fourth Euler substitution. In his original treatise, Leonhard Euler used two substitutions which are sufficient to cover all cases.
Databáze:	Directory of Open Access Journals
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