Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 7: Transcendental Functions - Section 7.7 - Hyperbolic Functions - Exercises 7.7 - Page 431: 67

Answer

$a.\quad \sinh^{-1}\sqrt{3}$ $b.\quad \ln(\sqrt{3}+2)$

Work Step by Step

$(a)$ Use table: "Integrals leading to inverse hyperbolic functions" 1. $\displaystyle \quad \int\frac{du}{\sqrt{a^{2}+u^{2}}}=\sinh^{-1}(\frac{u}{a})+C, \quad a\gt 0$ Here, a=2, $u(x)=x$. $ \displaystyle \int_{0}^{2\sqrt{3}}\frac{dx}{\sqrt{2^{2}+x^{2}}}=\left[ \sinh^{-1}(\frac{x}{2}) \right]_{0}^{-2\sqrt{3}}$ $=\sinh^{-1}\sqrt{3}-\sinh^{-1}0\qquad $... ( $\sinh 0=0 )$ $=\sinh^{-1}\sqrt{3}$ $(b)$ Using the formulas in the box above these exercises, $\sinh^{-1}x=\ln(x+\sqrt{x^{2}+1})$ ,$\quad -\infty \lt x \lt \infty$ $\sinh^{-1}\sqrt{3}=\ln(\sqrt{3}+\sqrt{3+1})=\ln(\sqrt{3}+2)$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.