## Calculus with Applications (10th Edition)

Published by Pearson

# Chapter 13 - The Trigonometric Functions - 13.3 Integrals of Trigonometric Functions - 13.3 Exercises - Page 697: 17

#### Answer

$$- 3\ln \left| {\cos \frac{1}{3}x} \right| + C$$

#### Work Step by Step

\eqalign{ & \int {\tan \frac{1}{3}x} dx \cr & {\text{set }}u = \frac{1}{3}x{\text{ then }}\frac{{du}}{{dx}} = \frac{1}{3},\,\,\,\,\,\,\,\,3du = dx \cr & {\text{write the integrand in terms of }}u \cr & \int {\tan \frac{1}{3}x} dx = \int {\tan u\left( {3du} \right)} \cr & {\text{use multiple constant rule}} \cr & = 3\int {\tan udu} \cr & {\text{ using the Basic Trigonometric integral }}\int {\tan x} dx = - \ln \left| {\cos x} \right| + C{\text{ }}\left( {{\text{see page 694}}} \right) \cr & = 3\left( { - \ln \left| {\cos u} \right|} \right) + C \cr & = - 3\ln \left| {\cos u} \right| + C \cr & {\text{write in terms of }}x \cr & = - 3\ln \left| {\cos \frac{1}{3}x} \right| + C \cr}

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