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: 8

Answer

\[ = \frac{{{{\sec }^{12}}x}}{{12}} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {{{\sec }^{12}}x\tan xdx} \hfill \\ \hfill \\ split\,\,\,\,\,{\sec ^{12}}x = {\sec ^{11}}x\sec x \hfill \\ \hfill \\ therefore \hfill \\ \hfill \\ \int_{}^{} {{{\sec }^{12}}\tan x\,dx} = \int_{}^{} {{{\sec }^{11}}x\sec x\tan xdx} \hfill \\ \hfill \\ set\,\,u = \sec \,x \to \,du = \sec x\tan xdx \hfill \\ \hfill \\ then \hfill \\ \hfill \\ \int_{}^{} {{{\sec }^{11}}x\sec x\tan xdx} = \int {{u^{11}}du} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ \frac{{{u^{12}}}}{{12}} + C \hfill \\ \hfill \\ substitute\,\,back\,\,u = \sec x \hfill \\ \hfill \\ = \frac{{{{\sec }^{12}}x}}{{12}} + C \hfill \\ \end{gathered} \]
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