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

Method of Partial Fractions

Known Integrals

NOTE: You should know the following indefinite integrals.

These will be seen with great frequency when using the Method of Partial Fractions:

• ${\displaystyle \int \frac{1}{ax+b}dx=\frac{1}{a}\ln |ax+b|+C}$$\displaystyle \int{\frac{1}{ax+b}\,dx}=\frac{1}{a}\ln{\left|ax+b\right|}+C$

• ${\displaystyle \int \frac{1}{x^2+a^2}dx=\frac{1}{a}{\tan }^{-1}\left(\frac{x}{a}\right)+C}$$\displaystyle \int{\frac{1}{x^2+a^2}\,dx}=\frac{1}{a}\tan^{-1}{\left(\frac{x}{a}\right)}+C$

Investigation 06

Use FTC-I to verify the above known integrals.

Investigation 07

Evaluate the following integrals:

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

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

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

4. ${\displaystyle \int \frac{1}{6x+1}dx}$$\displaystyle \int{\frac{1}{6x+1}\,dx}$

5. ${\displaystyle \int \frac{1}{7-4x}dx}$$\displaystyle \int{\frac{1}{7-4x}\,dx}$

Investigation 08

Evaluate the following integrals:

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

2. ${\displaystyle \int \frac{1}{x^2+9}dx}$$\displaystyle \int{\frac{1}{x^2+9}\,dx}$

3. ${\displaystyle \int \frac{1}{2x^2+18}dx}$$\displaystyle \int{\frac{1}{2x^2+18}\,dx}$

4. ${\displaystyle \int \frac{1}{5x^2+1}dx}$$\displaystyle \int{\frac{1}{5x^2+1}\,dx}$

5. ${\displaystyle \int \frac{1}{3x^2+7}dx}$$\displaystyle \int{\frac{1}{3x^2+7}\,dx}$

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Created: Wednesday, 08 September 2021 - 06:24 (EDT) Last Modified: Wednesday, 17 August 2022 - 09:46 (EDT)