Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.2 Integration by Parts - 7.2 Exercises: 14

Answer

\[ = \theta \tan \theta + \ln \left| {\cos \theta } \right| + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\theta {{\sec }^2}\theta d\theta } \hfill \\ \hfill \\ set\,\,\,the\,\,substitution \hfill \\ \hfill \\ u = \theta \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \to \,\,\,\,\,du = d\theta \hfill \\ dv = {\sec ^2}\theta dt\,\,\, \to \,\,\,\,v = \tan \theta \hfill \\ \hfill \\ use\,\,uv - \int_{}^{} {vdu} \hfill \\ \hfill \\ {\text{replacing the values }}{\text{in the equation}} \hfill \\ \hfill \\ = \theta \tan \theta - \int_{}^{} {\tan \theta \,\,d\theta } \hfill \\ or \hfill \\ = \theta \tan \theta + \int_{}^{} {\frac{{ - \sin \theta }}{{\cos \theta }}} d\theta \hfill \\ integrate \hfill \\ \hfill \\ = \theta \tan \theta + \left( {\ln \left| {\cos \theta } \right|} \right) + C \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ = \theta \tan \theta + \ln \left| {\cos \theta } \right| + C \hfill \\ \end{gathered} \]
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