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

Definite Integrals

Investigation 17

Evaluate the following definite integral:

1. ${\displaystyle {\int }_1^3\frac{1}{x\sqrt{3+x^2}}dx}$$\displaystyle \int_{1}^{3}{\frac{1}{x\sqrt{3+x^2}}\,dx}$

Use Technology to Verify Definite Integrals

Mathematica

NOTE: Mathematica is used here to verify definite integrals. On all Assessments, you will be permitted to use technology to verify definite integrals; however, you must show all Calculus II related work when using Trigonometric Substitution.

1
(* Integrate 1/(x sqrt(3 + x^2)) from x=1 to x=3 *)
2
Integrate[1/(x Sqrt[3 + x^2]), {x, 1, 3}]

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. All Mathematica code must be in black except for any variables or parameters.

3. Mathematica uses $\text{Sqrt[ ]}$$\text{Sqrt[\ \ ]}$ for $\sqrt{}$$\sqrt{\,\,\,}$.

4. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

Challenge Problem 18

Evaluate the following definite integral:

1. ${\displaystyle {\int }_0^{\ln (3)}\frac{e^x}{\sqrt{e^{2x}+4}}dx}$$\displaystyle \int_0^{\ln{(3)}}{\frac{e^x}{\sqrt{e^{2x}+4}}\,dx}$

HINT: Rewrite $e^{2x}=(e^x{)}^2$$e^{2x}=(e^x)^2$, then use $u$$u$-Substitution, then use Trigonometric Substitution.

Check Your Work

Use Technology to Verify Definite Integrals

Mathematica

NOTE: Mathematica is used here to verify definite integrals. On all Assessments, you will be permitted to use technology to verify definite integrals; however, you must show all Calculus II related work when using Trigonometric Substitution.

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2
1
(* Integrate e^x/sqrt(e^(2x) + 4) from x=0 to x=ln(3) *)
2
Integrate[E^x/Sqrt[E^(2x) + 4], {x, 0, Log[3]}]

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. All Mathematica code must be in black except for any variables or parameters.

3. Mathematica uses $\text{Sqrt[ ]}$$\text{Sqrt[\ \ ]}$ for $\sqrt{}$$\sqrt{\,\,\,}$; $\text{E}\wedge x$$\text{E}\!\wedge\!x$, $\text{Log[3]}$$\text{Log[3]}$ for $\ln (3)$$\ln{(3)}$.

4. To execute code (including comment codes), press and hold the SHIFT key and press the ENTER key.

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: Sunday, 12 June 2022 - 20:55 (EDT)