Partial Fraction

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

Combining Rational Expressions

Investigation 04

Find a common denominator and combine the following difference as a single rational expression:

${\displaystyle \frac{1}{2}\left(\frac{1}{x+3}\right)-\frac{1}{2}\left(\frac{1}{x+5}\right)}$$\displaystyle \frac{1}{2}\left(\frac{1}{x+3}\right)-\frac{1}{2}\left(\frac{1}{x+5}\right)$

Use Technology to Rewrite Expressions

NOTE: On Assessments, you may use of technology to rewrite expressions without showing any work.

Mathematica

1
(* Combine 1/2 1/(x+3) - 1/2 1/(x+5) *)
2
FullSimplify[1/2*1/(x+3)-1/2*1/(x+5)]

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: http://www.jjw3.com/TECH_Common_Functions.pdf.

2. FullSimplify[ ] has one argument:

   1. The expression to be simplified.

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

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

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

   3. Click on local

   4. Read how to use the FullSimplify[ ] 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.

Definition: Partial Fractions

Since ${\displaystyle \frac{1}{x^2+8x+15}}$$\displaystyle \frac{1}{x^2+8x+15}$ can be decomposed into a sum of ${\displaystyle \frac{1}{2}\left(\frac{1}{x+3}\right)}$$\displaystyle \frac{1}{2}\left(\frac{1}{x+3}\right)$ and ${\displaystyle -\frac{1}{2}\left(\frac{1}{x+5}\right)}$$\displaystyle -\frac{1}{2}\left(\frac{1}{x+5}\right)$, then

${\displaystyle \frac{1}{2}\left(\frac{1}{x+3}\right)}$$\displaystyle \frac{1}{2}\left(\frac{1}{x+3}\right)$ and ${\displaystyle -\frac{1}{2}\left(\frac{1}{x+5}\right)}$$\displaystyle -\frac{1}{2}\left(\frac{1}{x+5}\right)$ are partial fractions of ${\displaystyle \frac{1}{x^2+8x+15}}$$\displaystyle \frac{1}{x^2+8x+15}$.

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Created: Monday, 6 September 2020 13:33 EDT Last Modified: Wednesday, 08 September 2021 - 05:58 (EDT)