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

Special Integrals

You should remember the following integrals:

• ${\displaystyle \int \tan (x)dx=\ln |\sec (x)|+C}$$\displaystyle \int{\tan{(x)}\,dx}=\ln\left|\sec{(x)}\right|+C$

• ${\displaystyle \int \sec (x)dx=\ln |\sec (x)+\tan (x)|+C}$$\displaystyle \int{\sec{(x)}\,dx}=\ln\left|\sec{(x)}+\tan{(x)}\right|+C$

Investigation 10

Use FTC-I to verify the above special integrals.

NOTE: Mathematica will not return the expected antiderivative for these integrals.

Derivation of the Special Integrals

NOTE: You will NOT be assessed on the derivation of these integrals.

Integral of Tangent Function

$\int \tan (x)dx=\int {\displaystyle \frac{\sin (x)}{\cos (x)}}dx$$\int{\tan{(x)}\,dx} = \int{\dfrac{\sin{(x)}}{\cos{(x)}}\,dx}$

Let $u=\cos (x)$$u=\cos{(x)}$; $du=-\sin (x)$$du=-\sin{(x)}$

$\Rightarrow -\int {\displaystyle \frac{1}{u}}du$$\Rightarrow -\int{\dfrac{1}{u}\,du}$

$\Rightarrow -\ln |u|+C$$\Rightarrow -\ln{|\,u\,|}+C$

$\Rightarrow -\ln |\cos (x)|+C$$\Rightarrow -\ln{|\,\cos{(x)}\,|}+C$

$\Rightarrow \ln |(\cos (x){)}^{-1}|+C$$\Rightarrow \ln{|\,(\cos{(x)})^{-1}\,|}+C$

$\Rightarrow \ln |\sec (x)|+C$$\Rightarrow \ln{|\,\sec{(x)}\,|}+C$

NOTE: Any of the last three lines above are acceptable as the antiderivative of $\tan (x)$$\tan{(x)}$.

Integral of Secant Function

$\int \sec (x)dx=\int {\displaystyle \frac{\sec (x)}{1}}\left({\displaystyle \frac{\sec (x)+\tan (x)}{\sec (x)+\tan (x)}}\right)dx$$\int{\sec{(x)}\,dx} =\int{\dfrac{\sec{(x)}}{1}\left(\dfrac{\sec{(x)} + \tan{(x)}}{\sec{(x)} + \tan{(x)}}\right)\,dx}$

$\Rightarrow \int {\displaystyle \frac{{\sec }^2(x)+\sec (x)\tan (x)}{\sec (x)+\tan (x)}}dx$$\Rightarrow\int{\dfrac{\sec^2{(x)} + \sec{(x)}\tan{(x)}}{\sec{(x)} + \tan{(x)}}\,dx}$

Let $u=\tan (x)+\sec (x)$$u=\tan{(x)}+\sec{(x)}$; $du={\sec }^2(x)+\sec (x)\tan (x)$$du=\sec^2{(x)}+\sec{(x)}\tan{(x)}$

$\Rightarrow \int {\displaystyle \frac{1}{u}}du$$\Rightarrow\int{\dfrac{1}{u}\,du}$

$\Rightarrow \ln |u|+C$$\Rightarrow\ln{|\,u\,|}+C$

$\Rightarrow \ln |\tan (x)+\sec (x)|+C$$\Rightarrow\ln{|\,\tan{(x)} + \sec{(x)}\,|}+C$

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Last Modified: Tuesday, 1 September 2020 04:01 EDT