Chapter 7 - Integration Techniques - 7.3 Trigonometric Integrals - 7.3 Exercises: 28

$$- \ln \left| {\cos \theta } \right| - \frac{{{{\tan }^2}\theta }}{2} + C$$

Work Step by Step

\eqalign{ & \int {{{\tan }^3}\theta d\theta } \cr & {\text{split off }}\tan \theta \cr & = \int {{{\tan }^2}\theta \tan \theta d\theta } \cr & {\text{pythagorean identity}} \cr & = \int {\left( {1 - {{\sec }^2}\theta } \right)} \tan \theta d\theta \cr & {\text{Split }} \cr & = \int {\tan \theta } d\theta - \int {{{\sec }^2}\theta } \tan \theta d\theta \cr & {\text{evaluate the integral}} \cr & = - \ln \left| {\cos \theta } \right| - \frac{{{{\tan }^2}\theta }}{2} + C \cr}

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