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

Before we consider the Trigonometric Substitution Method, we need to review some essential Algebra and Trigonometry concepts.

Reducing Radical Expressions

Investigation 03

Consider the expression $\sqrt{4-x^2}$$\sqrt{4-x^2}$.

1. Explain why $\sqrt{4-x^2}\ne \sqrt{(2-x)(2-x)}$$\sqrt{4-x^2}\ne \sqrt{(2-x)(2-x)}$.

2. Explain why $\sqrt{4-x^2}\ne (2-x)$$\sqrt{4-x^2}\ne (2-x)$.

Check Your Work

Use Technology to Verify if Two Algebraic Expressions are Equivalent

Mathematica

NOTE: Mathematica is used here to identify patterns; you will NOT need to use $===$$===$ on any Assessment.

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(* Verify: sqrt(4 - x^2) is not equal to sqrt((2-x)(2-x)) *)
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Sqrt[4-x^2] === Sqrt[(2-x)(2-x)]
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(* Verify: sqrt(4 - x^2) is not equal to (2-x) *)
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Sqrt[4-x^2] === (2-x)

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: http://www.jjw3.com/TECH_Common_Functions.pdf.

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. Mathematica uses $===$$===$ as a Boolean operator, i.e., to determine if two expressions are equivalent or not. The Boolean operator returns either True or False.

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

Activity 04

Consider the expression $\sqrt{4-x^2}$$\sqrt{4-x^2}$. Substitute $x=2\sin (\theta )$$x=2\sin{(\theta)}$. Substitute $x$$x$ into the expression and simplify. In other words, simplify ${\displaystyle \sqrt{4-[2\sin (\theta ){]}^2}}$$\displaystyle \sqrt{4-[2\sin{(\theta)}]^2}$.

Check Your Work

Use Technology to Rewrite Trigonometric Expression

Mathematica

NOTE: Mathematica is used here to identify patterns; you will NOT need to use PowerExpand on any Assessment.

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(* Simplify: sqrt(4 - (2 sin(x))^2) *)
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FullSimplify[Sqrt[4 - (2 Sin[x])^2]]//PowerExpand

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: http://www.jjw3.com/TECH_Common_Functions.pdf.

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{Sin[ ]}$$\text{Sin[\ \ ]}$ for $\sin (x)$$\sin{(x)}$, etc.

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

Activity 05

Consider the expression $\sqrt{9+x^2}$$\sqrt{9+x^2}$. Substitute $x=3\tan (\theta )$$x=3\tan{(\theta)}$. Substitute $x$$x$ into the expression and simplify. In other words, simplify ${\displaystyle \sqrt{9+[3\tan (\theta ){]}^2}}$$\displaystyle \sqrt{9+[3\tan{(\theta)}]^2}$.

Check Your Work

Use Technology to Rewrite Trigonometric Expression

Mathematica

NOTE: Mathematica is used here to identify patterns; you will NOT need to use PowerExpand on any Assessment.

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(* Simplify: sqrt(9 + (3 tan(x))^2) *)
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FullSimplify[Sqrt[9 + (3 Tan[x])^2]]//PowerExpand

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: http://www.jjw3.com/TECH_Common_Functions.pdf.

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{Tan[ ]}$$\text{Tan[\ \ ]}$ for $\tan (x)$$\tan{(x)}$, etc.

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

Activity 06

Consider the expression $(x^2-16{)}^{3/2}$$(x^2-16)^{3/2}$. Substitute $x=4\sec (\theta )$$x=4\sec{(\theta)}$ into the expression and simplify. In other words, simplify ${\displaystyle ([4\sec (\theta ){]}^2-16{)}^{3/2}}$$\displaystyle ([4\sec{(\theta)}]^2-16)^{3/2}$.

Check Your Work

Use Technology to Rewrite Trigonometric Expression

Mathematica

NOTE: Mathematica is used here to identify patterns; you will NOT need to use PowerExpand on any Assessment.

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(* Simplify: ((4\sec(\theta))^2-16)^(3/2) *)
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FullSimplify[((4 Sec[x])^2 - 16)^(3/2)]//PowerExpand

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: http://www.jjw3.com/TECH_Common_Functions.pdf.

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{Sec[ ]}$$\text{Sec[\ \ ]}$ for $\sec (x)$$\sec{(x)}$, etc.

4. Since the exponent has more than one factor, the exponent needs to be grouped with parens $($$($ and $)$$)$.

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

Investigation 07

What do you notice about the results from Activity 04, Activity 05, and Activity 06? Explain.

Activity 08

1. Evaluate $\sqrt{4}$$\sqrt{4}$.

2. Evaluate $\sqrt{5^2}$$\sqrt{5^2}$.

3. Evaluate $\sqrt{(-3{)}^2}$$\sqrt{(-3)^2}$.

Check Your Work

Use Technology to Rewrite Trigonometric Expression

Mathematica

NOTE: Mathematica is used here to identify patterns; you will NOT need to use PowerExpand on any Assessment.

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(* Evalaute: sqrt(4) *)
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FullSimplify[Sqrt[4]]//PowerExpand

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: http://www.jjw3.com/TECH_Common_Functions.pdf.

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.

Investigation 09

What do you notice about the results from Activity 08? Explain.

Activity 10

1. From Algebra and Trigonometry, we know $\sqrt{{\cos }^2(x)}=\cos (x)$$\sqrt{\cos^2{(x)}}=\cos{(x)}$.

2. From Algebra and Activity 08, we know $\sqrt{(\cos (x){)}^2}=\cos (x)\ge 0$$\sqrt{(\cos{(x)})^2}=\cos{(x)}\geq 0$.

3. Where [i.e., on what interval(s) of $x$$x$-values from $x=-\pi$$x=-\pi$ to $x=\pi$$x=\pi$] is $\cos (x)\ge 0$$\cos{(x)}\geq 0$?

4. Here is an embedded DESMOS graph to assist with the question in Activity 10-3:

Activity 11

1. From Algebra and Trigonometry, we know $\sqrt{(\tan (x){)}^2}=\tan (x)$$\sqrt{(\tan{(x)})^2}=\tan{(x)}$.

2. From Algebra and Activity 08, we know $\sqrt{(\tan (x){)}^2}=\tan (x)\ge 0$$\sqrt{(\tan{(x)})^2}=\tan{(x)}\geq 0$.

3. Where [i.e., on what interval(s) of $x$$x$-values from $x=0$$x=0$ to $x=2\pi$$x=2\pi$] is $\tan (x)\ge 0$$\tan{(x)}\geq 0$?

4. Here is an embedded DESMOS graph to assist with the question in Activity 11-3:

Activity 12

1. From Algebra and Trigonometry, we know $\sqrt{(\sec (x){)}^2}=\sec (x)$$\sqrt{(\sec{(x)})^2}=\sec{(x)}$.

2. From Algebra and Activity 08, we know $\sqrt{(\sec (x){)}^2}=\sec (x)\ge 0$$\sqrt{(\sec{(x)})^2}=\sec{(x)}\geq 0$.

3. Where [i.e., on what interval(s) of $x$$x$-values from $x=-\pi$$x=-\pi$ to $x=\pi$$x=\pi$] is $\sec (x)\ge 0$$\sec{(x)}\geq 0$?

4. Here is an embedded DESMOS graph to assist with the question in Activity 12-3:

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Created: Monday, 6 September 2020 13:33 EDT Last Modified: Wednesday, 01 September 2021 - 07:07 (EDT)