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: 33

Answer

\[ = \frac{{{{\sec }^3}x}}{3} + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\tan x{{\sec }^3}xdx} \hfill \\ \hfill \\ {\text{rewrite}}\,\,\,{\text{the}}\,\,{\text{integrand}} \hfill \\ \hfill \\ \int_{}^{} {{{\sec }^2}x\tan x\sec xdx} \hfill \\ \hfill \\ use\,\,\,{\sec ^2}x = {u^2}\,\,\,\,\,then\,\,\,\tan x\sec x\,dx = \,du \hfill \\ \hfill \\ \int_{}^{} {{{\sec }^2}x\tan x\sec xdx} = \int_{}^{} {{u^2}du} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = \frac{{{u^3}}}{3} + C \hfill \\ \hfill \\ substituting\,\,back\,\,u\, = \,\sec x \hfill \\ \hfill \\ = \frac{{{{\sec }^3}x}}{3} + C \hfill \\ \end{gathered} \]
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