Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.3 Trigonometric Integrals - 7.3 Exercises: 42

Answer

\[ = \frac{{{{\tan }^6}\theta }}{6} + \frac{{{{\tan }^8}\theta }}{8} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {{{\tan }^5}\theta \,{{\sec }^4}\theta \,d\theta } \hfill \\ \hfill \\ write\,\,{\tan ^5}\theta \,{\sec ^4}\theta {\text{ as }}\,\,{\tan ^5}\theta \,{\sec ^2}\,\theta \,{\sec ^2}\theta \hfill \\ \hfill \\ = \int_{}^{} {{{\tan }^5}\theta \,{{\sec }^2}\,\theta \,{{\sec }^2}\theta d\theta } \hfill \\ \hfill \\ use\,the\,identity\,\,{\sec ^2}\theta = 1 + {\tan ^2}\theta d\theta \hfill \\ \hfill \\ = \int_{}^{} {{{\tan }^5}\theta \,\left( {1 + {{\tan }^2}\theta } \right){{\sec }^2}\theta d\theta } \hfill \\ \hfill \\ multiply\,and\,distribute\, \hfill \\ \hfill \\ = \int_{}^{} {{{\tan }^5}\theta {{\sec }^2}\theta d\theta + \int_{}^{} {{{\tan }^7}\theta {{\sec }^2}\theta d\theta } } \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = \frac{{{{\tan }^6}\theta }}{6} + \frac{{{{\tan }^8}\theta }}{8} + C \hfill \\ \end{gathered} \]
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