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

1. If integrand contains ${\displaystyle {(a^2-x^2)}^{\frac{n}{2}}}$$\displaystyle \color{red}\left(a^2-x^2\right)^{\frac{n}{2}}$, $n\in \mathbb{Z}$$n\in\mathbb{Z}$ (i.e., for any integer $n$$n$)

   1. Let ${\displaystyle x=a\sin (\theta )}$$\displaystyle x=a\sin{(\theta)}$, ${\displaystyle -\frac{\pi }{2}\le \theta \le \frac{\pi }{2}}$$\displaystyle -\frac{\pi}{2}\leq \theta\leq\frac{\pi}{2}$

   2. ${\displaystyle dx=a\cos (\theta )d\theta }$$\displaystyle dx=a\cos{(\theta)}d\theta$

   3. Use the Pythagorean identity: ${\displaystyle a^2-a^2{\sin }^2(\theta )=a^2{\cos }^2(\theta )}$$\displaystyle a^2-a^2\sin^2{(\theta)}=a^2\cos^2{(\theta)}$

   4. Construct right triangle using: ${\displaystyle \sin (\theta )=\frac{x}{a}}$$\displaystyle \sin{(\theta)}=\frac{x}{a}$

2. If integrand contains ${\displaystyle {(x^2-a^2)}^{\frac{n}{2}}}$$\displaystyle \color{red}\left(x^2-a^2\right)^{\frac{n}{2}}$​, $n\in \mathbb{Z}$$n\in\mathbb{Z}$ (i.e., for any integer $n$$n$)

   1. Let ${\displaystyle x=a\sec (\theta )}$$\displaystyle x=a\sec{(\theta)}$, ${\displaystyle 0\le \theta <\frac{\pi }{2}}$$\displaystyle 0\leq \theta <\frac{\pi}{2}$, $\pi \le \theta <\frac{3\pi }{2}$$\pi\leq \theta <\frac{3\pi}{2}$

   2. ${\displaystyle dx=a\sec (\theta )\tan (\theta )d\theta }$$\displaystyle dx=a\sec{(\theta)}\tan{(\theta)}d\theta$

   3. Use the Pythagorean identity: ${\displaystyle a^2{\sec }^2(\theta )-a^2=a^2{\tan }^2(\theta )}$$\displaystyle a^2\sec^2{(\theta)}-a^2=a^2\tan^2{(\theta)}$

   4. Construct right triangle using: ${\displaystyle \sec (\theta )=\frac{x}{a}}$$\displaystyle \sec{(\theta)}=\frac{x}{a}$

3. If integrand contains ${\displaystyle {(a^2+x^2)}^{\frac{n}{2}}}$$\displaystyle \color{red}\left(a^2+x^2\right)^{\frac{n}{2}}$​, $n\in \mathbb{Z}$$n\in\mathbb{Z}$ (i.e., for any integer $n$$n$)

   1. Let ${\displaystyle x=a\tan (\theta )}$$\displaystyle x=a\tan{(\theta)}$, ${\displaystyle -\frac{\pi }{2}<\theta <\frac{\pi }{2}}$$\displaystyle -\frac{\pi}{2}< \theta <\frac{\pi}{2}$

   2. ${\displaystyle dx=a{\sec }^2(\theta )d\theta }$$\displaystyle dx=a\sec^2{(\theta)}d\theta$

   3. Use the Pythagorean identity: ${\displaystyle a^2+a^2{\tan }^2(\theta )=a^2{\sec }^2(\theta )}$$\displaystyle a^2+a^2\tan^2{(\theta)}=a^2\sec^2{(\theta)}$

   4. Construct right triangle using: ${\displaystyle \tan (\theta )=\frac{x}{a}}$$\displaystyle \tan{(\theta)}=\frac{x}{a}$

Investigation 13

Compare the angles shown Substitutions Section above with the respective angles identified in Activities 10–12 in ACT_05d_Trig_Sub_Important_Concepts.html.

Trigonometric Substitution Method

1. Identify which substitution is needed: for example, Let ${\displaystyle x=a\sin (\theta )}$$\displaystyle x=a\sin{(\theta)}$, ${\displaystyle -\frac{\pi }{2}\le \theta \le \frac{\pi }{2}}$$\displaystyle -\frac{\pi}{2}\leq \theta\leq\frac{\pi}{2}$.

2. Differentiate the substitution: for the above example, ${\displaystyle dx=a\cos (\theta )}$$\displaystyle dx=a\cos{(\theta)}$.

3. Replace all $x$$x$-variables with the substitution in the integrand. All variables remaining in the integrand must only be ${\displaystyle \theta }$$\displaystyle \theta$.

4. Replace $dx$$dx$ in the integrand.

5. Simplify the integral.

6. Integrate with respect to ${\displaystyle \theta }$$\displaystyle \theta$.

7. Substitute back into terms of ${\displaystyle x}$$\displaystyle x$ using the right triangle definitions (See PRE_05_Trig_Sub.html) of the substitution.

NOTE: This last step is an important step in integration using Trig-Sub. This step must be used to completely convert $\theta$$\theta$-variables into $x$$x$-variables.

Use Technology for Trigonometric Identities

NOTE: In Step 7 above, the constructed right triangle can only convert $\sin (\theta )$$\sin{(\theta)}$, $\cos (\theta )$$\cos{(\theta)}$, $\tan (\theta )$$\tan{(\theta)}$, etc. to $x$$x$-variables. If your antiderivative consists of $\sin (n\theta )$$\sin{(n\theta)}$, $\cos (n\theta )$$\cos{(n\theta)}$, $\tan (n\theta )$$\tan{(n\theta)}$, etc. where $n\ne 1$$n\ne 1$, then you need to use a Trigonometric Identity:

Mathematica

NOTE: Mathematica is used here to convert trigonometric ratios of a multiple of an angle, $n\theta$$n\theta$, to a trigonometric ratio of the angle, $\theta$$\theta$. On all Assessments, you will be permitted to use technology to convert trig ratios without showing any work.

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(* Find trig identity for cos(4t) *)
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TrigExpand[Cos[4t]]

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{Cos[x]}$$\text{Cos[x]}$ for $\cos (x)$$\cos{(x)}$.

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

Investigation 14

Evaluate the following indefinite integrals:

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

4. ${\displaystyle \int \frac{dx}{{(9+x^2)}^{\frac{3}{2}}}}$$\displaystyle \int{\frac{dx}{\left(9+x^2\right)^{\frac{3}{2}}}}$

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

6. ${\displaystyle \int \frac{\sqrt{x^2-16}}{x}dx}$$\displaystyle \int{\frac{\sqrt{x^2-16}}{x}\,dx}$

7. ${\displaystyle \int \frac{1}{\sqrt{5-x^2}}dx}$$\displaystyle \int{\frac{1}{\sqrt{5-x^2}}\,dx}$

Check Your Work

Use Technology to Verify Antiderivatives

Mathematica

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

xxxxxxxxxx
2
1
(* Integrate 1/(x^2 sqrt(x^2-1)) *)
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Integrate[1/(x^2 Sqrt[x^2 - 1]), 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: 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.

Homework

At this time, you should be able to complete the following assignments:

• Section 7.3: # 1, 3, 5, 13, 17, 19.

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Created: Monday, 6 September 2020 13:33 EDT Last Modified: Wednesday, 01 June 2022 - 16:51 (EDT)