Trigonometric Integrals

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

• Section 7.2 – Trigonometric Integrals

Expected Educational Results

• Objective 04–01: I can evaluate an indefinite trigonometric integral.

• Objective 04–02: I can evaluate a definite trigonometric 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 Integrals

Half- and Double-Angle Formulas

From Pre-Calculus, we know:

• ${\displaystyle {\cos }^2(x)=\frac{1+\cos (2x)}{2}}$$\displaystyle \cos^2{\!(x)}=\frac{1+\cos{\!(2x)}}{2}$

• ${\displaystyle {\sin }^2(x)=\frac{1-\cos (2x)}{2}}$$\displaystyle \sin^2{\!(x)}=\frac{1-\cos{(2x)}}{2}$

Example 02: Evaluate ${\displaystyle \int {\cos }^4(x)\text{d}x}$$\displaystyle\int{\cos^4{\!(x)}\,\text{d}x}$

Solution 02:

We discussed in a previous page that $u$$u$-Substitution is not valid here. Thus, we need to rewrite:

${\displaystyle {\cos }^4(x)={\cos }^2(x){\cos }^2(x)=\left(\frac{1+\cos (2x)}{2}\right)\left(\frac{1+\cos (2x)}{2}\right)=\frac{1+2\cos (2x)+{\cos }^2(2x)}{4}}$$\displaystyle \cos^4{\!(x)}=\cos^2{\!(x)}\cos^2{\!(x)}=\left(\frac{1+\cos{\!(2x)}}{2}\right)\left(\frac{1+\cos{\!(2x)}}{2}\right)=\frac{1+2\cos{\!(2x)}+\cos^2{\!(2x)}}{4}$

We will also need to rewrite: ${\cos }^2(2x)$$\color{red}\cos^2{\!(2x)}$

${\displaystyle \Rightarrow \frac{1+2\cos (2x)+{\cos }^2(2x)}{4}=\frac{1+2\cos (2x)+\frac{1+\cos (4x)}{2}}{4}}$$\displaystyle \Rightarrow\frac{1+2\cos{\!(2x)}+\cos^2{\!(2x)}}{4}=\frac{1+2\cos{\!(2x)}+{\color{red}\frac{1+\cos{\!(4x)}}{2}}}{4}$

Use algebra to rewrite:

${\displaystyle \frac{1+2\cos (2x)+\frac{1+\cos (4x)}{2}}{4}=\frac{3}{8}+\frac{1}{2}\cos (2x)+\frac{1}{8}\cos (4x)}$$\displaystyle \frac{1+2\cos{\!(2x)}+{\color{red}\frac{1+\cos{\!(4x)}}{2}}}{4}=\frac{3}{8}+\frac{1}{2}\cos{\!(2x)}+\frac{1}{8}\cos{\!(4x)}$

Thus,

${\displaystyle \int {\cos }^4(x)\text{d}x=\int (\frac{3}{8}+\frac{1}{2}\cos (2x)+\frac{1}{8}\cos (4x))dx}$$\displaystyle\int{\cos^4{\!(x)}\,\text{d}x}=\int{\left(\frac{3}{8}+\frac{1}{2}\cos{\!(2x)}+\frac{1}{8}\cos{\!(4x)}\right)\,dx}$

which is now integrable using $u$$u$-Substitution:

${\displaystyle \Rightarrow =\int (\frac{3}{8}+\frac{1}{2}\cos (2x)+\frac{1}{8}\cos (4x))dx=\frac{3}{8}x+\frac{1}{2}\frac{\sin (2x)}{2}+\frac{1}{8}\frac{\sin (4x)}{4}+C}$$\displaystyle\Rightarrow=\int{\left(\frac{3}{8}+\frac{1}{2}\cos{\!(2x)}+\frac{1}{8}\cos{\!(4x)}\right)\,dx}=\frac{3}{8}x+\frac{1}{2}\frac{\sin{\!(2x)}}{2}+\frac{1}{8}\frac{\sin{\!(4x)}}{4}+C$ ∎

Use Technology to Rewrite Using Algebra

NOTE: You may use Mathematica to rewrite integrals in which sine and cosine are both to even powers without showing any associated work.

Mathematica

1
(* Find expression equivalent to cos(x)^2 *)
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TrigReduce[Cos[x]^2]

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. The Mathematica syntax for ${\cos }^2(x)$$\cos^2{(x)}$ is $\text{Cos[x]}\wedge 2$$\text{Cos[x]}\!\wedge\!2$​, etc.

3. Remember, correct Mathematica code will be all black except for variables.

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

Python

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 Python syntax is located at: https://www.jjw3.com/Common_Python_Code.html.

2. All code is included above with comment codes that explain the executable code.

3. The Python syntax for ${\cos }^2(x)$$\cos^2{(x)}$​​​ is $\text{cos(x)**2}$$\text{cos(x)**2}$​​​; etc.

4. Python requires $\ast$$*$ for explicit multiplication; $\ast \ast$$**$​ for exponents.

5. To execute code, press the “View the result” button:

Investigation 09

Use the trigonometric identities to evaluate the following integrals.

Use Mathematica or Python to first rewrite these expressions.

1. ${\displaystyle \int {\cos }^2(x)dx}$$\displaystyle \int{\cos^{2}{\!(x)}\,dx}$

2. ${\displaystyle \int {\sin }^2(x)dx}$$\displaystyle \int{\sin^{2}{\!(x)}\,dx}$

3. ${\displaystyle \int {\sin }^2(x){\cos }^2(x)dx}$$\displaystyle \int{\sin^{2}{\!(x)}\cos^2{\!(x)}\,dx}$

4. ${\displaystyle \int {\sin }^4(x)dx}$$\displaystyle \int{\sin^{4}{\!(x)}\,dx}$

5. ${\displaystyle \int {\sin }^2(x){\cos }^4(x)dx}$$\displaystyle \int{\sin^{2}{\!(x)}\cos^4{\!(x)}\,dx}$

Use Technology to Verify Antiderivatives

Mathematica

NOTE: Mathematica is used here to verify antiderivatives. You must show that you understand the Calculus concept of the antiderivative.

xxxxxxxxxx
2
1
(* Example from Investigation 08-3: Integrate sin(x)^2 cos(x)^2 *)
2
Integrate[Sin[x]^2*Cos[x]^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: https://www.jjw3.com/Common_Mathematica_Code.html.

2. The Mathematica syntax for ${\cos }^2(x)$$\cos^2{(x)}$ is $\text{Cos[x]}\wedge 2$$\text{Cos[x]}\!\wedge\!2$​, etc.

3. Integrate[ ] for indefinite integrals has two arguments:

   1. The integrand, i.e., function of the definite integral;

   2. $x$$x$​, where $x$$x$​ is the variable used in the integrand.

4. Remember, correct Mathematica code will be all black except for variables.

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

6. Mathematica does not show $+C$$+C$ for antiderivatives. You will need to remember to include $+C$$+C$ in your answers.

7. Mathematica may return a result in a different, but equivalent, form than your answer. In these cases, use FTC-I to check your work.

Python

NOTE: Python is used here to verify definite integrals. On all Assessments, you must show that you understand the Calculus concept of using $u$$u$-Substitution to evaluate an indefinite integral.

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 Python syntax is located at: https://www.jjw3.com/Common_Python_Code.html.

2. All code is included above with comment codes that explain the executable code.

3. integrate( ) for indefinite integrals has two arguments:

   1. The integrand, i.e., function of the definite integral;

   2. $x$$x$​, where $x$$x$​​ is the variable used in the integrand.

4. You only need to change the function in the integrate( ) command.

5. The Python syntax for ${\cos }^2(x)$$\cos^2{(x)}$​​​ is $\text{cos(x)**2}$$\text{cos(x)**2}$​​​; etc.

6. Python requires $\ast$$*$ for explicit multiplication; $\ast \ast$$**$​ for exponents.

7. Python does not return the antiderivative with $+C$$+C$. You need to use $+C$$+C$ for all steps that are antiderivatives.

8. To execute code, press the “View the result” button: ​​

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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: Tuesday, 1 September 2020 03:43 EDT Last Modified: Tuesday, 16 August 2022 - 21:19 (EDT)