University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 7 - Section 7.3 - Hyperbolic Functions - Exercises - Page 418: 45


$7 \ln ( \cosh \dfrac{x}{7}) +C= 7 \ln |e^{x/7}+e^{-x/7}| +C$

Work Step by Step

Given: $\int \tanh (\dfrac{x}{7}) dx$ Plug in $\dfrac{x}{7}=a$ and $dx= 7 da$ Thus, $\int \tanh (\dfrac{x}{7}) dx= 7 \int \tanh a da =7 \int \dfrac{\sinh a}{\cosh a} da $ Now, plug in $\cosh a =u \implies \sinh a da =du$ Thus, we have: $=7 \int \dfrac{dt}{t}$ or, $= 7 \ln |u| +C$ or, $= 7 \ln |\cosh \dfrac{x}{7}| +C$ $= 7 \ln |\frac{e^{x/7}+e^{-x/7}}{2}| +C$ $= 7 \ln |e^{x/7}+e^{-x/7}| +C$
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