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On Geometric Interpretation of Euler’s Substitutions
Version 1
: Received: 27 September 2023 / Approved: 28 September 2023 / Online: 28 September 2023 (09:59:35 CEST)
A peer-reviewed article of this Preprint also exists.
Cieśliński, J.L.; Jurgielewicz, M. On Geometric Interpretations of Euler’s Substitutions. Symmetry 2023, 15, 1932. Cieśliński, J.L.; Jurgielewicz, M. On Geometric Interpretations of Euler’s Substitutions. Symmetry 2023, 15, 1932.
Abstract
We consider a classial case of irrational integrals containing a square root of a quadratic polynomial. It is well known that they can be expressed in terms of elementary functions by one of three Euler’s substitutions. It is less known that the Euler substittutions have a beautiful 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 the fourth Euler substitution. By the way, it is not clear who was the first to attribute these three substitutions to Euler. In his original treatise Leonhard Euler uses two substitutions which are sufficient to cover all cases.
Keywords
integral calculus; irrational integrals; conics; rational parameterization; fourth Euler’s substitution
Subject
Computer Science and Mathematics, Analysis
Copyright: This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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