Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 7 - Techniques of Integration - 7.1 Integration by Parts - 7.1 Exercises - Page 516: 25

Answer

$$\frac{e^4+3}{2e^2}$$

Work Step by Step

Given $$ \int_{0}^{2}y\sinh ydy$$ Let \begin{align*} u&=y\ \ \ \ \ \ \ \ \ \ \ \ \ \ dv=\sinh ydy\\ u&= dy\ \ \ \ \ \ \ \ \ \ \ \ \ dv=\cosh y \end{align*} Then using integration by parts \begin{align*} \int_{0}^{2} y\sinh y dy &=uv-\int vdu\\ &= y\cosh y\bigg|_{0}^{2}-\int_{0}^{2} \cosh ydy\\ &=y\cosh y-\sinh y \bigg|_{0}^{2}\\ &=\frac{e^4+3}{2e^2} \end{align*}
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