Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.2 Integration by Parts - 7.2 Exercises: 8

Answer

\[ = - \frac{x}{2}\cos 2x + \frac{1}{4}\sin 2x\, + C\]

Work Step by Step

\[\begin{gathered} \int_{}^{} {x\sin \,2x\,dx} \hfill \\ \hfill \\ set\,\,\,the\,\,substitution \hfill \\ \hfill \\ u = x\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,then\,\,\,\,\,\,\,du = dx \hfill \\ du = \sin 2xdx\,\,\,\,\,\,\,then\,\,\,v = - \frac{1}{2}\cos \,2x \hfill \\ \hfill \\ use\,\,uv - \int_{}^{} {vdu} \hfill \\ \hfill \\ {\text{replacing the values }}{\text{in the equation}} \hfill \\ \hfill \\ = - \frac{x}{2}\cos 2x - \int_{}^{} {\,\left( { - \frac{1}{2}\cos 2x} \right)} dx \hfill \\ \hfill \\ = - \frac{x}{2}\cos 2x + \frac{1}{4}\int_{}^{} {\,\left( {\cos 2x} \right)\left( 2 \right)} dx \hfill \\ \hfill \\ \operatorname{int} egrating \hfill \\ \hfill \\ = - \frac{x}{2}\cos 2x + \frac{1}{4}\sin 2x\, + C \hfill \\ \end{gathered} \]
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