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 the Method of Partial Fractions

Investigation 03 – Reprise

I have used algebra to rewrite the following integrals.

Determine if trigonometric substitution is a valid method for evaluating the following integrals. Explain.

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

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

3. ${\displaystyle \int \frac{x+2}{x^2-2x+1}dx=\int \frac{x+2}{{(x-1)}^2}dx}$$\displaystyle\int{\frac{x+2}{x^2-2x+1}\,dx} = \int{\frac{x+2}{\left(x-1\right)^2}\,dx}$ HINT: Do you see any of the following: $x^2-a^2$$x^2-a^2$, $a^2-x^2$$a^2-x^2$, or $x^2+a^2$$x^2+a^2$ (where a is a constant)?

4. ${\displaystyle \int \frac{x}{x^3-1}dx=\int \frac{x}{(x-1)({(x+\frac{1}{2})}^2+\frac{3}{4})}dx}$$\displaystyle\int{\frac{x}{x^3-1}\,dx} = \int{\frac{x}{(x-1)\left(\left(x + \frac{1}{2}\right)^2+\frac{3}{4}\right)}\,dx}$ HINT: What do you notice about the factor $x-1$$x-1$ in the denominator?

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: Wednesday, 08 September 2021 - 05:54 (EDT)