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

Additional Substitutions

Investigation 04

For each of the following integrals, the correct $u$$u$-Substitution is given. Complete only the first step for $u$$u$-Substitution, then cancel and common factors in the integral.

Explain why an additional step IS required after an initial $u$$u$-substitution for the following integrals:

1. ${\displaystyle \int {\sin }^2(\theta ){\cos }^3(\theta )d\theta }$$\displaystyle\int{\sin^2{\!(\theta)}\cos^{3}{\!(\theta)}\,d\theta}$ [$u=\sin (x)$$u=\sin{(x)}$]

2. ${\displaystyle \int {\sin }^4(\theta ){\cos }^3(\theta )d\theta }$$\displaystyle\int{\sin^{4}{\!(\theta)}\cos^{3}{\!(\theta)}\,d\theta}$ [$u=\sin (x)$$u=\sin{(x)}$]

3. ${\displaystyle \int {\sin }^3(\theta ){\cos }^3(\theta )d\theta }$$\displaystyle\int{\sin^{3}{\!(\theta)}\cos^{3}{\!(\theta)}\,d\theta}$ [$u=\sin (x)$$u=\sin{(x)}$]

4. ${\displaystyle \int {\sin }^3(\theta ){\cos }^4(\theta )d\theta }$$\displaystyle\int{\sin^{3}{\!(\theta)}\cos^{4}{\!(\theta)}\,d\theta}$ [$u=\cos (x)$$u=\cos{(x)}$]

5. ${\displaystyle \int {\sin }^5(x){\cos }^4(x)dx}$$\displaystyle\int{\sin^{5}{\!(x)}\cos^{4}{\!(x)}\,dx}$ [$u=\cos (x)$$u=\cos{(x)}$]

6. ${\displaystyle \int {\sin }^3(x){\cos }^2(x)dx}$$\displaystyle\int{\sin^{3}{\!(x)}\cos^{2}{\!(x)}\,dx}$ [$u=\cos (x)$$u=\cos{(x)}$]

7. ${\displaystyle \int {\sec }^4(t){\tan }^3(t)dt}$$\displaystyle\int{\sec^{4}{\!(t)}\tan^3{\!(t)}\,dt}$ [$u=\tan (x)$$u=\tan{(x)}$]

8. ${\displaystyle \int {\sec }^4(t){\tan }^2(t)dt}$$\displaystyle\int{\sec^{4}{\!(t)}\tan^{2}{\!(t)}\,dt}$ [$u=\tan (x)$$u=\tan{(x)}$]

9. ${\displaystyle \int {\sec }^6(x){\tan }^3(x)dx}$$\displaystyle\int{\sec^{6}{\!(x)}\tan^{3}{\!(x)}\,dx}$ [$u=\tan (x)$$u=\tan{(x)}$]

10. ${\displaystyle \int {\sec }^6(x){\tan }^3(x)dx}$$\displaystyle\int{\sec^{6}{\!(x)}\tan^{3}{\!(x)}\,dx}$ [$u=\sec (x)$$u=\sec{(x)}$]

11. ${\displaystyle \int {\sec }^2(x){\tan }^5(x)dx}$$\displaystyle\int{\sec^{2}{\!(x)}\tan^{5}{\!(x)}\,dx}$ [$u=\sec (x)$$u=\sec{(x)}$]

12. ${\displaystyle \int {\sec }^3(t){\tan }^3(t)dt}$$\displaystyle\int{\sec^{3}{\!(t)}\tan^{3}{\!(t)}\,dt}$ [$u=\sec (x)$$u=\sec{(x)}$]

13. ${\displaystyle \int {\csc }^4(x){\cot }^5(x)dx}$$\displaystyle\int{\csc^{4}{\!(x)}\cot^{5}{\!(x)}\,dx}$ [$u=\cot (x)$$u=\cot{(x)}$]

14. ${\displaystyle \int {\csc }^4(x){\cot }^5(x)dx}$$\displaystyle\int{\csc^{4}{\!(x)}\cot^{5}{\!(x)}\,dx}$ [$u=\csc (x)$$u=\csc{(x)}$]

15. ${\displaystyle \int {\csc }^2(x){\cot }^3(x)dx}$$\displaystyle\int{\csc^{2}{\!(x)}\cot^{3}{\!(x)}\,dx}$ [$u=\csc (x)$$u=\csc{(x)}$]

Investigation 05

1. What pattern(s) do you notice after the initial $u$$u$-substitution and cancelations in the integrals in Investigation 04? Explain. Try to find a single pattern that works for all of the above integrals.

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Created: Tuesday, 1 September 2020 03:23 EDT Last Modified: Wednesday, 01 June 2022 - 15:22 (EDT)