Integration in finite terms: the Liouville principle and the Ostrowski method

• Autor(a) principal: Silva, Allan Kenedy Santos
• Data de Publicação: 2024
• Tipo de documento: Artigo
• Idioma: por
• Título da fonte: Remat (Bento Gonçalves)
• Texto Completo: https://periodicos.ifrs.edu.br/index.php/REMAT/article/view/6556
• topic: integração elementar; integração em finitos termos; princípio de Liouville; teorema de Liouville; teorema de Ostrowski; elementary integration; integration in finite terms; Liouville principle; Liouville theorem; Ostrowski theorem; integración elemental; integración en términos finitos; principio de Liouville; teorema de Liouville; el teorema de Ostrowski

Since the beginnings of Differential and Integral Calculus, many mathematicians have dedicated years of their lives to the development of this subject. They improved several techniques for computing the integrals of various classes of functions, but there were some of them that they could not calculate in terms of elementary functions (functions expressed by a finite number of polynomials, radicals, exponentials, logarithms, and trigonometric functions, using a finite amount of algebraic operations and function compositions). A question then arose about whether such integrals were in fact elementary. This led to the French mathematician Joseph Liouville developing a theory of integration in finite terms. In this paper, we presenting Liouville's brilliant reasoning and a generalization proposed by Ukrainian mathematician Alexander Ostrowski. Besides that, we will also be displaying possible applications of their results in the calculation of some integrals.