Unit 7A

Integrals Part 1

Lesson 7: Integration by Partial Fractions

Integration by Partial Fractions

In Unit 1 Lesson 6 , a rational function was defined as a function written as a fraction where both numerator and denominator are polynomials and which can be written in the form «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mi»f«/mi»«mi mathvariant=¨normal¨»(«/mi»«mi»x«/mi»«mi mathvariant=¨normal¨»)«/mi»«mo»=«/mo»«mfrac»«mrow»«mi»P«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«mrow»«mi»Q«/mi»«mfenced»«mi»x«/mi»«/mfenced»«/mrow»«/mfrac»«mi mathvariant=¨normal¨»,«/mi»«mo»§#160;«/mo»«mi»Q«/mi»«mfenced»«mi»x«/mi»«/mfenced»«mo»§#8800;«/mo»«mn»0«/mn»«/math». Rational functions are comprised of one of two types of rational expressions: proper and improper. When the degree of the numerator is less than the degree of the denominator, it is a proper rational expression. When the degree of the numerator is greater than the degree of the denominator, it is an improper rational expression.

For example, the rational expression «math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mn»8«/mn»«mi»x«/mi»«mo»+«/mo»«mn»7«/mn»«/mrow»«mrow»«msup»«mi»x«/mi»«mn»2«/mn»«/msup»«mo»+«/mo»«mi»x«/mi»«mo»§#8722;«/mo»«mn»2«/mn»«/mrow»«/mfrac»«/math» is a proper rational expression because the degree in the numerator is less than the degree in the denominator.

A rational function comprised of a proper rational expression can more easily be integrated when it is written as the sum or difference of two simpler proper rational expressions. The challenge is in finding the two simpler expressions that are equivalent to the original.

Begin by considering how to add rational expressions.

Find a common denominator.


To resolve a rational expression into two fractions, this process needs to be reversed. The method used to express a single fraction as a sum of two or more fractions is called the method of partial fractions.