Integration of Rational Functions

Abstract

Preliminary Remarks The integrals of rational functions form the simplest class of integrals; they are included in a first course in calculus. Yet some problems associated with the integration of rational function have connections with the deeper aspects of algebra and of analysis. Examples are the factorization of polynomials and the evaluation of beta integrals. These problems have challenged mathematical minds as great as Newton, Johann Bernoulli, de Moivre, Euler, Gauss, and Hermite; indeed, they have their puzzles for us even today. For example, can a rational function be integrated without factorizing the denominator of the function? Newton was the first mathematician to explicitly define and systematically attack the problem of integrating rational and algebraic functions. Of course, mathematicians before Newton had integrated some specific rational functions, necessary for their work. The Kerala mathematicians found the series for arctangent; N. Mercator and Hudde worked out the series for the logarithm. But Newton's work was made possible by his discovery, sometime in mid-1665, of the inverse relation between the derivative and the integral. At that time, he constructed tables, extending to some pages, of functions that could be integrated because they were derivatives of functions already explicitly or implicitly defined. He extended his tables by means of substitution or, equivalently, by use of the chain rule for derivatives. He further developed the tables by an application of the product rule for derivatives, or integration by parts, in his October 1666 tract on fluxions.