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

Reminder

So far, I have suggested that you should follow the following general steps when integrating a function:

1. Use FTC-I, if possible;

2. Rewrite, then use FTC-I, if possible;

3. Use $u$$u$-sub; you may still need to rewrite the integral until all $x$$x$-variables are converted into $u$$u$-variables.

Practice 13

Evaluate the following integrals.

1. ${\displaystyle {\int }_{\pi /6}^{\pi /4}{\sec }^4(x){\tan }^3(x)dx}$$\displaystyle \int_{\pi/6}^{\pi/4}{\sec^{4}{\!(x)}\tan^{3}{\!(x)}\,dx}$

2. ${\displaystyle {\int }_{\pi /6}^{\pi /3}{\sec }^3(x){\tan }^3(x)dx}$$\displaystyle \int_{\pi/6}^{\pi/3}{\sec^{3}{\!(x)}\tan^{3}{\!(x)}\,dx}$

3. ${\displaystyle {\int }_{\pi /4}^{\pi /3}{\sec }^4(x){\tan }^2(x)dx}$$\displaystyle \int_{\pi/4}^{\pi/3}{\sec^{4}{\!(x)}\tan^{2}{\!(x)}\,dx}$

4. ${\displaystyle {\int }_0^{\pi /4}{\tan }^2(x)dx}$$\displaystyle \int_{0}^{\pi/4}{\tan^{2}{\!(x)}\,dx}$

5. ${\displaystyle {\int }_{\pi /4}^{\pi /4}{\sec }^2(x)dx}$$\displaystyle \int_{\pi/4}^{\pi/4}{\sec^{2}{\!(x)}\,dx}$

6. ${\displaystyle {\int }_0^{3\pi /4}{\sin }^4(x){\cos }^5(x)dx}$$\displaystyle \int_{0}^{3\pi/4}{\sin^{4}{\!(x)}\cos^{5}{\!(x)}\,dx}$

7. ${\displaystyle {\int }_{\pi /3}^{\pi }{\sin }^4(x){\cos }^2(x)dx}$$\displaystyle \int_{\pi/3}^{\pi}{\sin^{4}{\!(x)}\cos^{2}{\!(x)}\,dx}$

8. ${\displaystyle {\int }_{\pi /4}^{\pi /2}{\sin }^5(x)\cos (x)dx}$$\displaystyle \int_{\pi/4}^{\pi/2}{\sin^{5}{\!(x)}\cos{(x)}\,dx}$

9. ${\displaystyle {\int }_0^{\pi }{\sin }^3(x){\cos }^3(x)dx}$$\displaystyle \int_{0}^{\pi}{\sin^3{\!(x)}\cos^3{\!(x)}\,dx}$

10. ${\displaystyle {\int }_{-\pi }^{\pi }{\cos }^2(x)dx}$$\displaystyle \int_{-\pi}^{\pi}{\cos^{2}{\!(x)}\,dx}$

Use Technology to Verify Definite Integrals

Mathematica

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

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(* Example from Investigation 02: Integrate sin(x)^3 cos(x) *)
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Integrate[Sin[x]^3*Cos[x], 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 ${\sec }^2(x)$$\sec^2{(x)}$ is $\text{Sec[x]}\wedge 2$$\text{Sec[x]}\!\wedge\!2$​, etc.

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

   1. The integrand, i.e., function of the indefinite 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 indefinite integrals. On all Assessments, you must show that you understand the Calculus concept of evaluating 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 indefinite 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. Remember to use parens $($$\left(\right.$ and $)$$\left.\right)$​ to group multiple terms in the numerator or denominator of a rational expression.

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

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

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Created: Tuesday, 16 August 2022 - 21:51 (EDT)