# Chapter 13 - The Trigonometric Functions - 13.3 Integrals of Trigonometric Functions - 13.3 Exercises - Page 697: 20

$$- 2\ln \left| {\cos {{\left( {\frac{x}{4}} \right)}^2}} \right| + C$$

#### Work Step by Step

\eqalign{ & \int {\frac{x}{4}\tan {{\left( {\frac{x}{4}} \right)}^2}} dx \cr & or \cr & \int {\frac{x}{4}\tan \left( {\frac{{{x^2}}}{{16}}} \right)} dx \cr & {\text{set }}u = \frac{{{x^2}}}{{16}}{\text{ then }}\frac{{du}}{{dx}} = \frac{x}{8},\,\,\,\,\,\,\,\,\frac{{8du}}{x} = dx \cr & {\text{write the integrand in terms of }}u \cr & \int {\frac{x}{4}\tan \left( {\frac{{{x^2}}}{{16}}} \right)} dx = \int {\frac{x}{4}\tan u} \left( {\frac{{8du}}{x}} \right) \cr & {\text{cancel the common factors}} \cr & = \int {\tan u} \left( {2du} \right) \cr & {\text{use multiple constant rule}} \cr & = 2\int {\tan u} du \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 & = - 2\ln \left| {\cos u} \right| + C \cr & {\text{write in terms of }}x \cr & = - 2\ln \left| {\cos \frac{{{x^2}}}{{16}}} \right| + C \cr & or \cr & = - 2\ln \left| {\cos {{\left( {\frac{x}{4}} \right)}^2}} \right| + C \cr}

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