Integration of Rational Functions 7.4 Integration of Rational Functions By Partial Fractions

Example 1 This would be a lot easier if we could re-write it as two separate terms. Multiply by the common denominator. Set like-terms equal to each other. Solve two equations with two unknowns.

This technique is called Partial Fractions Solve two equations with two unknowns.

The short-cut for this type of problem is called the Heaviside Method, after English engineer Oliver Heaviside. Multiply by the common denominator. Let x = - 1 Let x = 3

Method of Partial Fractions If the fraction is improper, use long division to rewrite it as a sum of a polynomial and a proper fraction. If the fraction is proper, factor the denominator completely and write it as a sum of fractions as follows: a) For each linear factors (ax+b)n, the decomposition must have the form: b) ) For each irreducible quadratic factors (ax2+bx+c)n, the decomposition must have the form:

Example 2 Repeated roots: we must use two terms for partial fractions.

Example 3 If the degree of the numerator is higher than the degree of the denominator, use long division first. (from example one)

Example 4 irreducible quadratic factor repeated root

Examples