Teorema da Liouville: uma aplicação na integração de funções.

Abstract

This study we will present some of the history of calculus, emphasizing the development of the Fundamental Theorem of Calculus- (TFC) one of the most important theorems of this area, being a essential tool addressed in courses of higher levels, of paramount importance and of essential application. Some important concepts related to elementary functions along their properties, definitions, and integration results will be presented. In addition, investigated the problem of how to solve integrals that cannot be calculated by applying TFC. Therefore, the main objective of this work is to know if, given a function f, its primitive can be expressed or not "in elementary terms". Thus, it is sought to give a precise meaning to the notion of integration in these terms, where we will present two theorems, Chebyshev’s theorem and especially Liouville’s theorem, that establish a practical path, making it possible to know whether some functions are expressed in elementary terms or not. Thus, although this problem is not widely explored in the calculus books, it is of great relevance to be a study.