Trigonometric Substitution

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

• Section 7.3 – Trigonometric Substitution

Expected Educational Results

• Objective 05–01: I can use a trigonometric substitution to evaluate an indefinite integral.

• Objective 05–02: I can use a trigonometric substitution 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.

Trigonometric Substitution

Trigonometric Substitutions

Investigation 01

Explain why $u$$u$-substitution is NOT valid method for the following integrals:

1. ${\displaystyle \int x^4\sqrt{1-x^2}dx}$$\displaystyle\int{x^4\sqrt{1-x^2}\,dx}$

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

3. ${\displaystyle \int \frac{1}{x^2\sqrt{x^2-1}}dx}$$\displaystyle\int{\frac{1}{x^2\sqrt{x^2-1}}\,dx}$

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: Monday, 6 September 2020 13:33 EDT Last Modified: Wednesday, 01 September 2021 - 06:50 (EDT)