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

Mathematical Syntax

We usually write the square of $\sin (x)$$\sin{(x)}$ as ${\sin }^2(x)$$\sin^2{(x)}$ which is equivalent to the composition ${(\sin (x))}^2$$\left(\sin{(x)} \right)^2$, $\sin {(x)}^2$$\sin{(x)}^2$, $\text{Sin[x]}\wedge 2$$\text{Sin[x]}\!\wedge\!2$ [Mathematica], or $\sin \left(x\right)\ast \ast 2$$\sin(x)\!*\!*2$ [Python] and is NOT equivalent to $\sin x^2$$\sin{x}^2$​.

NOTE: ${\sin }^{-1}(x)\ne {\displaystyle \frac{1}{\sin (x)}}$$\sin^{-1}{(x)}\ne\dfrac{1}{\sin{(x)}}$; i.e., ${\sin }^{-1}(x)$$\sin^{-1}{(x)}$ is NOT a composition function.

Investigation 01

Explain why $u$$u$-substitution IS a valid method for the following integrals:

1. ${\displaystyle \int {\sin }^3(\theta )\cos (\theta )d\theta }$$\displaystyle\int{\sin^{3}{\!(\theta)}\cos{(\theta)}\,d\theta}$

2. ${\displaystyle \int {\sec }^2(t)\tan (t)dt}$$\displaystyle\int{\sec^{2}{\!(t)}\tan{(t)}\,dt}$

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

4. ${\displaystyle \int {\sec }^2(x){\tan }^3(x)dx}$$\displaystyle\int{\sec^{2}{\!(x)}\tan^{3}{\!(x)}\,dx}$ [HINT: Consider $u=\tan (x)$$u=\tan{(x)}$]

5. ${\displaystyle \int {\csc }^2(x){\cot }^2(x)dx}$$\displaystyle\int{\csc^{2}{\!(x)}\cot^{2}{\!(x)}\,dx}$ [HINT: Consider $u=\cot (x)$$u=\cot{(x)}$]

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Last Modified: Monday, 18 January 2021 10:01 EDT