Substitution Rule

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 5.5 – The Substitution Rule

Expected Educational Results

• Objective 02–01: I can use a substitution to evaluate an indefinite integral using FTC-I.

• Objective 02–02: I can use a substitution to evaluate a definite integral.

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.

Substitution Rule

Identifying When the Substitution Rule is Not Needed​​

Investigation 18

What pattern(s) do you notice about the following indefinite integrals? Explain why the Substitution Rule is not required?

1. ${\displaystyle \int {\sec }^2(x)dx}$$\displaystyle \int{\sec^2{\!(x)}\,dx}$

2. ${\displaystyle \int {\tan }^2(x)dx}$$\displaystyle \int{\tan^2{\!(x)}\,dx}$

3. ${\displaystyle \int (e^x{)}^{2}dx}$$\displaystyle \int{(e^x)^2\,dx}$

Additional Special Cases

Here are two additional cases where the Substitution Rule is required; however, we will not be able to integrate the following until we learn the Integration By Parts method.

1. ${\displaystyle \int e^{\sqrt{x}}dx}$$\displaystyle \int{e^{\sqrt{x}}\,dx}$

2. ${\displaystyle \int \sin (e^x)dx}$$\displaystyle \int{\sin{(e^x)}\,dx}$

Investigation 19

What pattern(s) do you notice about the above special cases that suggest the Substitution Rule is required?

More Special Cases

Here are two additional cases where the Substitution Rule is not valid despite a composition of two functions; we will not be able to integrate the following until we learn more about integrating trigonometric functions.

1. ${\displaystyle \int {\cos }^2(x)dx}$$\displaystyle \int{\cos^2{\!(x)}\,dx}$

2. ${\displaystyle \int {\sin }^2(x)dx}$$\displaystyle \int{\sin^2{\!(x)}\,dx}$

Investigation 20

Summarize your understanding of the Substitution Rule by answering and explaining:

1. When is the Substitution Rule valid?

2. When is the Substitution Rule not valid?

3. How to choose the function that will be substituted with $u$$u$​.

4. How to verify that you chose the correct $u$$u$ substitution.

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Created: Tuesday, 16 August 2022 - 10:49 (EDT)