Calculus (3rd Edition)

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

Chapter 7 - Exponential Functions - 7.9 Hyperbolic Functions - Exercises - Page 384: 62


$$\cosh ^{-1} t =\ln (t+\sqrt{t^{2}-1})$$

Work Step by Step

Let $ t=\cosh x$. Since \begin{align*} \cosh^2 x-\sinh^2 x&=1\\ \sinh x&=\sqrt{\cosh^2 x-1}\\ &= \sqrt{t^2-1} \end{align*} and \begin{align*} \sinh x+\cosh x&=\frac{e^{x}-e^{-x}}{2}+\frac{e^{x}+e^{-x}}{2}\\ &=e^{x} \end{align*} Then \begin{align*} \cosh ^{-1} t&=x\\ &=\ln (\sinh x+\cosh x)\\ &=\ln (t+\sqrt{t^{2}-1}) \end{align*}
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