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: 54

Answer

\[ = 0\]

Work Step by Step

\[\begin{gathered} \int_{ - \pi /4}^{\pi /4} {{{\tan }^3}x{{\sec }^2}xdx} \hfill \\ \hfill \\ use\,\,{\text{ }}\int {{x^n}dx = \frac{{{x^{n + 1}}}}{{n + 1}} + C} \hfill \\ \hfill \\ = \,\,\left[ {\frac{{{{\tan }^4}x}}{4}} \right]_{ - \frac{\pi }{4}}^{\frac{\pi }{4}} \hfill \\ \hfill \\ evaluate\,\,the\,limits \hfill \\ \hfill \\ = \frac{{{{\tan }^4}\,\left( {\frac{\pi }{4}} \right)}}{4} - \frac{{{{\tan }^4}\,\,\,\left( { - \frac{\pi }{4}} \right)}}{4} \hfill \\ \hfill \\ simplify \hfill \\ \hfill \\ = \frac{{\,{{\left( 1 \right)}^4}}}{4} - \frac{{\,{{\left( { - 1} \right)}^4}}}{4} \hfill \\ \hfill \\ \hfill \\ = \frac{1}{4} - \frac{1}{4} = 0 \hfill \\ \end{gathered} \]
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