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

Need for Completing the Square with Rational Functions

Example 04: Consider ${\displaystyle \int \frac{3}{x^2-x+2}dx}$$\displaystyle \int{\frac{3}{x^2-x+2}\,dx}$.

Solution 04:

$x^2-x+2$$x^2-x+2$ is not factorable over the Reals.

By Completing the Square,

${\displaystyle x^2-x+2={(x-\frac{1}{2})}^2+\frac{7}{4}}$$\displaystyle x^2-x+2 = \left(x-\frac{1}{2}\right)^2 + \frac{7}{4}$

So,

${\displaystyle \int \frac{3}{x^2-x+2}dx=\int \frac{3}{{(x-\frac{1}{2})}^2+\frac{7}{4}}dx}$$\displaystyle \int{\frac{3}{x^2-x+2}\,dx}=\int{\frac{3}{\left(x-\frac{1}{2}\right)^2 + \frac{7}{4}}\,dx}$

By known integral formulas:

${\displaystyle \int \frac{3}{{(x-\frac{1}{2})}^2+\frac{7}{4}}dx=\frac{3}{\sqrt{\frac{7}{4}}}{\tan }^{-1}\left(\frac{x-\frac{1}{2}}{\sqrt{\frac{7}{4}}}\right)+C}$$\displaystyle \int{\frac{3}{\left(x-\frac{1}{2}\right)^2 + \frac{7}{4}}\,dx} = \frac{3}{\sqrt{\frac{7}{4}}} \tan^{-1}{\left(\frac{x-\frac{1}{2}}{\sqrt{\frac{7}{4}}}\right)}+C$

Investigation 11

Integrate the following functions.

1. ${\displaystyle \int \frac{1}{x^2-2x+4}dx}$$\displaystyle \int{\frac{1}{x^2-2x+4}\,dx}$

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

3. ${\displaystyle \int \frac{1}{x^2+3x-3}dx}$$\displaystyle \int{\frac{1}{x^2+3x-3}\,dx}$

4. ${\displaystyle \int \frac{1}{2x^2+x-3}dx}$$\displaystyle \int{\frac{1}{2x^2+x-3}\,dx}$

Use Technology to Factor

Mathematica

NOTE: You are permitted to use Mathematica to algebraically simplify expressions without showing any work.

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(* Example 04 *)
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(* Factor x^2-x+2 *)
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Factor[x^2-x+2]

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. Factor[ ] has one argument:

   1. The function to be factored, if possible.

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

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

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

   3. Click on local

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

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

5. If Mathematica returns the same expression, then the expression is not factorable.

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

Use Technology to Complete the Square

Mathematica

NOTE: You are permitted to use Mathematica to complete the square of an expression without showing any work.

​x
1
(* Verify x^2 - x + 2 is not factorable *)
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Factor[x^2-x+2]
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(* Complete the square: x^2 - x + 2 *)
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ResourceFunction["CompleteSquare"][x^2-x+2, x]

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

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

Check Your Work

Use Technology to Verify Indefinite Integrals

Mathematica

NOTE: The use of technology in this activity is to verify, i.e., check, your answers. You must show work on assessments to receive credit for all Calculus II work.

xxxxxxxxxx
2
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(* Example 04: Integrate 3/(x^2-x+2) *)
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Integrate[3/(x^2-x+2), x]

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. Integrate[ ] has two arguments:

   1. The function in the integrand;

   2. The independent variable.

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

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

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

   3. Click on local

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

4. You may need parens, $($$(\,$ and $)$$\,)$, to group multiple terms in the numerator and denominator.

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

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

7. Mathematica does not show $+C$$+C$ for antiderivatives. You will need to remember to include $+C$$+C$ in your answers.

8. Mathematica may return a result in a different, but equivalent, form than your answer. In these cases, use FTC-I to check your work.

Homework

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

• Section 7.4: # 9, 23.

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 - 05:29 (EDT)