Partial Fractions

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 7.4 – Integration of Rational Functions by Partial Fractions

Expected Educational Results

• Objective 06–01: I can use partial fractions to evaluate an indefinite integral.

• Objective 06–02: I can use partial fractions to evaluate a definite integral using FTC-II.

Bloom’s Taxonomy

A modern version of Bloom’s Taxonomy is included here to recognize various different levels of understanding and to encourage you to work towards higher-order understanding (those at the top of the pyramid). All Objectives, Investigations, Activities, etc. are color-coded with the level of understanding.

Partial Fractions

Need for Polynomial Division for Integrating Rational Functions

Let ${\displaystyle f(x)=\frac{P(x)}{Q(x)}}$$\displaystyle f(x)=\frac{P(x)}{Q(x)}$ be a rational function where $deg(P)\ge deg(Q)$$deg(P)\geq deg(Q)$, then polynomial division [or synthetic division] is required.

Investigation 12

Integrate the following functions.

1. ${\displaystyle \int \frac{x}{x+2}dx}$$\displaystyle \int{\frac{x}{x+2}\,dx}$

2. ${\displaystyle \int \frac{x^2+3}{x^2+4}dx}$$\displaystyle \int{\frac{x^2+3}{x^2+4}\,dx}$

3. ${\displaystyle \int \frac{x^2+3x-3}{x-3}dx}$$\displaystyle \int{\frac{x^2+3x-3}{x-3}\,dx}$

4. ${\displaystyle \int \frac{x^3-3x^2+x+2}{x^2-x}dx}$$\displaystyle \int{\frac{x^3-3x^2+x+2}{x^2-x}\,dx}$

Use Technology to Perform Polynomial Division

Mathematica

NOTE: You are permitted to use Mathematica to perform polynomial division without showing any work.

1
(* Investigation 09-4: Integrate (x^3-3x^2+x+2)/(x^2-x) *)
2
PolynomialQuotientRemainder[x^3-3x^2+x+2, x^2-x, x]

Mathematica Output: {x-2,2-x}

The quotient is $x-2$$\color{red}x-2$; the remainder is $2-x$$\color{blue}2-x$. Thus,

the output refers to the equivalence: ${\displaystyle {\displaystyle \frac{x^3-3x^2+x+2}{x^2-x}}=x-2+{\displaystyle \frac{2-x}{x^2-x}}}$$\displaystyle \dfrac{x^3-3x^2+x+2}{{\color{orange}x^2-x}} = {\color{red}x - 2} + \dfrac{{\color{blue}2 - x}}{{\color{orange}x^2 - x}}$

NOTE: You will need to decompose the rational expression into partial fractions.

Warnings:

1. Be very careful with the syntax. Syntax is the set of rules on how to write computer code. Every software program has its own unique syntax. Some basic Mathematica syntax is located at: https://www.jjw3.com/Common_Mathematica_Code.html.

2. PolynomialQuotientRemainder[ ] has three arguments:

   1. The dividend, i.e., the numerator of the rational expression;

   2. The divisor, i.e., the denominator of the rational expression;

   3. The independent variable.

3. For help on using the PolynomialQuotientRemainder[ ] function:

   1. In Mathematica, execute the code: $\text{?PolynomialQuotientRemainder}$$\text{?PolynomialQuotientRemainder}$

   2. Click on $\vee$$\or$ near the bottom-left of output

   3. Click on local

   4. Read how to use the PolynomialQuotientRemainder[ ] function – you will be able to copy-paste code.

4. Remember, correct Mathematica code will be all black except for variables.

5. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Homework

At this time, you should be able to complete the following assignments:

• Section 7.4: # 7.

CC BY-NC-SA 4.0

This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License [http://creativecommons.org/licenses/by-nc-sa/4.0/].

Created: Monday, 6 September 2020 13:33 EDT Last Modified: Wednesday, 15 September 2021 - 06:03 (EDT)