Topic: Integration Techniques

Subtopic: Partial Fractions

**Overview**

The process of decomposition of a rational function into a sum of partial fractions was covered in a pre-calculus class. For example the rational function `(2x-11)/(x^2+x-2)` can be factored `(2x-11)/[(x+2)(x-1)]` , expanded `A/(x+2)+B/(x-1)`, the A and B found, thus providing the final decomposition into the partial fractions, `(2x-11)/[(x+2)(x-1)] = 5/(x+2)-3/(x-1)`. The method of partial fractions is used in calculus to rewrite an integral of a rational function as a sum of integrals of simpler fractions. `int(2x-11)/(x^2+x-2)dx` becomes `int5/(x+2)dx +int(-3)/(x-1)dx` which are both easily integrable. Partial fractions ease the integration process when the function is rational.

**Objectives**

By the end of this topic you should know and be prepared to be tested on:

- 8.4.1 Perform partial fraction expansion on rational functions that may include linear factors, quadratic (or higher) factors, or repeated factors
- 8.4.2 Find the A, B, C, etc. constants in the partial fraction expansion by the "equating the coefficients" method and solving the related linear system (algebraically or electronically)
- 8.4.3 Find the A, B, C, etc. constants in the partial fraction expansion by solving for the constants algebraically via the "nifty x" method
- 8.4.4 Use partial fraction decomposition to rewrite an integral of a rational function as a sum of two (or more) integrals which can then be evaluated
- 8.4.5 use long division when necessary to simplify the rational function in an integral so that the method of partial fractions can be applied to the integral of the remainder over the dividend
- 8.4.6 Integrate rational functions expanded as partial fractions including those that involve the natural logarithm rule (`int(1/u)du=ln|u|+C`) and/or arctangent function

**Terminology**

Terms you should be able to define: partial fraction expansion, partial fraction decomposition, linear factors, quadratic factors, repeated factors, "equating the coefficients" method, "nifty x" method of solving a system during PF decomposition

**Mini-Lectures and Examples**

STUDY: Integration Techniques - Partial Fractions

**Supplemental Resources (recommended)**

It is a good idea to review how to perform decomposition of partial fractions from a pre-calculus course before integrating by the partial fraction method. Here is a link to my lesson on partial fractions from a college algebra course to help you review if you need it: Prof. Keely's Review of Partial Fraction Decomposition from Pre-Calculus

If you need supplemental tutorial videos with examples reviewing partial fractions from pre-calculus go to James Sousa's MathIsPower4U >> Algebra and search for topic: "Performing Partial Fraction Decomposition". Or better yet go straight to his MathIsPoswer4U >> Calc II course and search for topic: "Integration Using Partial Fractions" which incorporates precalc decomposition steps with each integation.

**Supplemental Resources (optional)**

Video: Partial Fraction Expansions, Selwyn Hollis's Video Calculus

Lesson: Partial Fraction Decomposition, Dale Hoffman's Contemporary Calculus

rev. 2021-02-06