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: 29

Answer

\[ = 4{\tan ^5}x - \frac{{20{{\tan }^3}x}}{3} + 20\tan x - 20x + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {20\,{{\tan }^6}xdx} \hfill \\ \hfill \\ write\,\,\,{\tan ^6}x\,\,as\,{\tan ^4}x{\tan ^2}x \hfill \\ \hfill \\ = 20\int_{}^{} {{{\tan }^4}x{{\tan }^2}xdx} \hfill \\ \hfill \\ Use\,\,the\,identity\,\,{\tan ^2}x = {\sec ^2}x - 1 \hfill \\ \hfill \\ = 20\int_{}^{} {{{\tan }^4}x\,\left( {{{\sec }^2}x - 1} \right)dx} \hfill \\ \hfill \\ multiply\, \hfill \\ \hfill \\ = 20\int_{}^{} {{{\tan }^4}x{{\sec }^2}xdx - 20\int_{}^{} {{{\tan }^4}xdx} } \hfill \\ \hfill \\ = 20\int_{}^{} {{{\tan }^4}{{\sec }^2}xdx} - 20\int_{}^{} {{{\tan }^2}x{{\tan }^2}xdx} \hfill \\ \hfill \\ Use\,\,\,the\,\,\,identity\,\,\,\,{\tan ^2}x = {\sec ^2}x - 1 \hfill \\ \hfill \\ = 20\int_{}^{} {{{\tan }^4}x{{\sec }^2}xdx} - 20\int_{}^{} {{{\tan }^2}x\,\left( {{{\sec }^2}x - 1} \right)dx} \hfill \\ \hfill \\ = 20\int_{}^{} {{{\tan }^4}x{{\sec }^2}xdx - 20\int_{}^{} {{{\tan }^2}x{{\sec }^2}xdx + 20\int_{}^{} {{{\tan }^2}xdx} } } \hfill \\ \hfill \\ = 20\int_{}^{} {{{\tan }^4}x{{\sec }^2}xdx} - 20\int_{}^{} {{{\tan }^2}x{{\sec }^2}xdx} + 20\int_{}^{} {\,\left( {{{\sec }^2}x - 1} \right)dx} \hfill \\ \hfill \\ integrate \hfill \\ \hfill \\ = 20\,\left( {\frac{{{{\tan }^5}x}}{5}} \right) - 20\,\left( {\frac{{{{\tan }^3}x}}{3}} \right) + 20\tan x - 20x + C \hfill \\ \hfill \\ = 4{\tan ^5}x - \frac{{20{{\tan }^3}x}}{3} + 20\tan x - 20x + C \hfill \\ \end{gathered} \]
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