Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.4 Integrals Involving Hyperbolic and Inverse Hyperbolic Functions - Exercises - Page 415: 19


$$\frac{1}{5} \sinh ^{-1}\left(\frac{5 x}{4}\right)+C $$

Work Step by Step

\begin{aligned} \int \frac{1}{\sqrt{16+25 x^{2}}} d x &=\int \frac{1}{\sqrt{16\left(1+\frac{25}{16} x^{2}\right)}} d x \\ &=\int \frac{1}{4 \sqrt{1+\left(\frac{5 x}{4}\right)^{2}}} d x \\ &=\frac{A}{5} \int \frac{\frac{5}{4}}{\sqrt{1+\left(\frac{5 x}{4}\right)^{2}}} d x \\ &=\frac{1}{5} \int \frac{\frac{5}{4}}{\sqrt{1+\left(\frac{5 x}{4}\right)^{2}}} d x \\ &=\frac{1}{5} \sinh ^{-1}\left(\frac{5 x}{4}\right)+C \end{aligned}
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