Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 7 - Integration Techniques - 7.6 Other Integration Strategies - 7.6 Exercises: 69

Answer

\[ = \frac{1}{9}{\tan ^3}\,\left( {3y} \right) - \frac{1}{3}\,\,\left[ {\tan \,\left( {3y} \right) - \,\left( {3y} \right)} \right] + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {{{\tan }^4}3y\,dy} \hfill \\ \hfill \\ Using\,\,the\,\,reduccion\,\,formula \hfill \\ \hfill \\ \int_{}^{} {{{\tan }^n}xdx} = \frac{{{{\tan }^{n - 1}}x}}{{n - 1}} - \int_{}^{} {{{\tan }^{n - 2}}xdx} \hfill \\ \hfill \\ set\,\,\,3y = u\,\,\,\,\,\,\,then\,\,\,\,\,\,dy = \frac{{du}}{3} \hfill \\ \hfill \\ \int_{}^{} {{{\tan }^4}\,\left( {3y} \right)} \,dy = \frac{1}{3}\int_{}^{} {{{\tan }^4}\,\left( u \right)\,du} \hfill \\ \hfill \\ integrating\,\,u\sin g\,\,the\,\,Reduction\,\,formula \hfill \\ \hfill \\ \int_{}^{} {{{\tan }^4}3y\,dy} = \frac{1}{3}\int_{}^{} {{{\tan }^4}\,\left( u \right)du} \hfill \\ \hfill \\ = \frac{1}{3}\,\,\left[ {\frac{{{{\tan }^3}u}}{3} - \int_{}^{} {{{\tan }^2}udu} } \right] \hfill \\ \hfill \\ {\text{Therefore}}{\text{,}} \hfill \\ \hfill \\ = \frac{1}{9}{\tan ^3}\,\left( {3y} \right) - \frac{1}{3}\,\,\left[ {\tan \,\left( {3y} \right) - \,\left( {3y} \right)} \right] + C \hfill \\ \end{gathered} \]
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