Calculus: Early Transcendentals (2nd Edition)

Published by Pearson

Chapter 7 - Integration Techniques - 7.3 Trigonometric Integrals - 7.3 Exercises: 38

Answer

$= - \ln \left| {\cos x} \right| - \frac{{{{\sin }^2}x}}{2} + C$

Work Step by Step

$\begin{gathered} \int_{}^{} {{{\sec }^{ - 2}}x{{\tan }^3}xdx} \hfill \\ \hfill \\ rewrite\,\,the\,\,{\text{integrand}} \hfill \\ \hfill \\ \,\,\int_{}^{} {{{\sec }^{ - 2}}x\tan x{{\tan }^2}xdx} \hfill \\ \hfill \\ use\,\,\,{\tan ^2}x = {\sec ^2}x - 1 \hfill \\ \hfill \\ = \int_{}^{} {{{\sec }^{ - 2}}x\tan x\,\left( {{{\sec }^2}x - 1} \right)dx} \hfill \\ \hfill \\ multiply \hfill \\ \hfill \\ = \int_{}^{} {\,\left( {\tan x - {{\sec }^{ - 2}}x\tan x} \right)dx} \hfill \\ \hfill \\ = \int_{}^{} {\,\left( {\tan x - \sin x\cos x} \right)dx} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = - \ln \left| {\cos x} \right| - \frac{{{{\sin }^2}x}}{2} + C \hfill \\ \end{gathered}$

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