University Calculus: Early Transcendentals (3rd Edition)

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

Chapter 8 - Section 8.5 - Integral Tables and Computer Algebra Systems - Exercises - Page 451: 35


$$\int\frac{dy}{y\sqrt{3+(\ln y)^2}}=\sinh^{-1}\frac{\ln y}{\sqrt3}+C$$

Work Step by Step

$$A=\int\frac{dy}{y\sqrt{3+(\ln y)^2}}$$ We set $u=\ln y$, which means $$du=\frac{dy}{y}$$ Therefore, $$A=\int\frac{du}{\sqrt{3+u^2}}$$ Use Formula 34, which states that $$\int\frac{dx}{\sqrt{a^2+x^2}}=\sinh^{-1}\frac{x}{a}+C$$ for $a=\sqrt3$. $$A=\sinh^{-1}\frac{u}{\sqrt3}+C$$ $$A=\sinh^{-1}\frac{\ln y}{\sqrt3}+C$$
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