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

Definite Integrals

Investigation 13

Integrate the following functions.

1. ${\displaystyle {\int }_1^{\pi /4}\frac{1}{x^3+4x}dx}$$\displaystyle \int_1^{\pi/4}{\frac{1}{x^3+4x}\,dx}$

2. ${\displaystyle {\int }_0^1\frac{x+3}{x^2-4}dx}$$\displaystyle \int_0^1{\frac{x+3}{x^2-4}\,dx}$

3. ${\displaystyle {\int }_0^3\frac{x+1}{(x^2+9{)}^2}}$$\displaystyle \int_0^3\frac{x+1}{(x^2+9)^2}$

4. ${\displaystyle {\int }_1^2\frac{x^2+x+2}{x^2+3x}}$$\displaystyle \int_1^2\frac{x^2+x+2}{x^2+3x}$

Check Your Work

Use Technology to Verify Definite Integrals

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.

1
(* Investigation 10-1: Integrate 1/(x^3+4x) from x=1 to x=pi/4 *)
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Integrate[1/(x^3+4x), {x, 1, Pi/4}]

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. $\{x,a,b\}$$\{x, a, b\}$, where

      1. $x$$x$ is the independent variable;

      2. $a$$a$ is the lower-limit of integration;

      3. $b$$b$ is the upper-limit of integration.

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.

Homework

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

• Section 7.4: # 11, 15, 19.

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Last Modified: Monday, 6 September 2020 13:33 EDT