Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.1 Basic Approaches - 7.1 Exercises: 45

Answer

\[ = \frac{2}{3}{\sin ^3}x + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {\sin x\,\sin \,2x\,dx} \hfill \\ \hfill \\ set\,\,\,\,\sin \,2x\, = 2\,\sin x\cos x \hfill \\ \hfill \\ then \hfill \\ \hfill \\ \int_{}^{} {\sin x\,\sin \,2x\,dx} = \int_{}^{} {\sin \,x\,\,\left( {2\,\sin x\,\,\cos x} \right)} \hfill \\ \hfill \\ multiply \hfill \\ \hfill \\ = 2\int_{}^{} {{{\sin }^2}x\,\,\,\cos x\,dx} \hfill \\ \hfill \\ set\,\,\sin \,x\, = t\,\, \to \,\cos x\,dx = dt \hfill \\ \hfill \\ 2\int_{}^{} {{{\sin }^2}x\,\,\,\cos x\,dx} = 2\int_{}^{} {{t^2}} dt\,\, \hfill \\ \hfill \\ in\,tegrate \hfill \\ \hfill \\ = 2\left( {\frac{{{t^3}}}{3}} \right) + C \hfill \\ \hfill \\ substituting\,back\,\,t = \,\sin x \hfill \\ \hfill \\ = \frac{2}{3}{\sin ^3}x + C \hfill \\ \end{gathered} \]
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