Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 8 - Techniques of Integration - 8.1 Integration by Parts - Exercises - Page 395: 30


\begin{aligned} \int x\ \sinh x\ d x= x \cosh x- \sinh x +C \end{aligned}

Work Step by Step

Given $$ \int x\ \ \sinh x\ d x $$ Use integration by parts: $$ u= x \Rightarrow du= dx $$ $$ dv=\sinh x \ dx \Rightarrow v= \cosh x $$ So, we get \begin{aligned} I&=\int x\ \sinh x\ d x\\ &=uv - \int vdu\\ &= x \cosh x-\int \cosh x dx\\ &= x \cosh x- \sinh x +C \end{aligned}
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