How to Integrate Rational Functions

Abstract

The increasing availability of computer algebra systems has raised questions about how traditional topics in calculus are to be taught. In this note we look at integration of rational functions and propose a different approach, which has the following advantages: i) it is easily implemented on a computer or calculator algebra system, ii) it allows the students to use the computer algebra system in a meaningful way, and avoids routine calculations by hand, iii) it provides the students with some understanding of the general methods computer algebra systems actually use to integrate rational functions. Rational function integration is important for itself and also because many integrals can be reduced to it by suitable substitutions, for example mnany trigonometric integrals and the so-called binomial integral [Subramaniam, Klambauer]. A rational function is traditionally integrated by expressing it in partial fractions form. This involves the following steps: (1) Factor the denominator into linear and irreducible quadratic factors. (2) Find the partial fraction decomposition. This involves solving a system of linear equations, with as many equations and unknowns as the degree of the polynomial in the denominator. (3) Integrate each partial fraction. Those involving a quadratic factor require a trigonometric substitution or a reduction formula. In the light of this recipe, consider the following integrals (of which the second and third are taken from our references):