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 Needed​​

Investigation 07

What pattern(s) do you notice about the following indefinite integrals?

1. ${\displaystyle \int \frac{{\tan }^{-1}(x)}{1+x^2}dx}$$\displaystyle \int{\frac{{\color{c8}\tan^{-1}{(x)}}}{{\color{c1}1+x^2}}\,{\color{c1}dx}}$

2. ${\displaystyle \int \frac{{\sin }^{-1}(x)}{\sqrt{1-x^2}}dx}$$\displaystyle \int{\frac{{\color{c8}\sin^{-1}{(x)}}}{{\sqrt{\color{c1}1-x^2}}}\,{\color{c1}dx}}$​

3. ${\displaystyle \int \frac{\ln (x)}{x}dx}$$\displaystyle \int{\frac{\ln{(x)}}{x}\,dx}$

4. ${\displaystyle \int \tan (x){\sec }^2(x)dx}$$\displaystyle \int{\tan{(x)}\sec^2{(x)}\,{\color{c1}dx}}$​

5. ${\displaystyle \int \sin (x)\cos (x)dx}$$\displaystyle \int{{\color{c8}\sin}{\color{c8}(x)}\,{\color{c1}\cos{(x)}}\,{\color{c1}dx}}$

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Created: Monday, 9 August 2021 18:16 EDT Last Modified: Monday, 15 August 2022 - 16:03 (EDT)