Calculus: Early Transcendentals (2nd Edition)

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

Chapter 5 - Integration - 5.5 Substitution Rule - 5.5 Exercises: 67

Answer

\[ = \frac{1}{7}{\sec ^7}x + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\sin x{{\sec }^8}xdx} \hfill \\ \hfill \\ rewrite \hfill \\ \hfill \\ \int_{}^{} {\sin x\left( {\frac{1}{{\cos x}}} \right)\sec x{{\sec }^6}xdx} \hfill \\ \hfill \\ = \int_{}^{} {\,\left( {\frac{{\sin x}}{{\cos x}}} \right)\,\left( {\sec x} \right)\,\left( {{{\sec }^6}x} \right)} dx \hfill \\ \hfill \\ = \int_{}^{} {{{\sec }^6}x\,\left( {\sec x} \right)\,\left( {\tan x} \right)dx} \hfill \\ \hfill \\ set\,\,u = \sec x\,\,then\,\,du = \sec x\tan xdx \hfill \\ \hfill \\ = \int {{u^6}du} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ \frac{{{u^7}}}{7} + C \hfill \\ \hfill \\ replace\,\,u\,\,with\,\,u = \sec x \hfill \\ \hfill \\ = \frac{1}{7}{\sec ^7}x + C \hfill \\ \end{gathered} \]
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