Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.6 Strategies for Integration - Exercises - Page 431: 26


$$\ln \left|\frac{\sqrt{x^{2}+16}}{4}+\frac{x}{4}\right|+C$$

Work Step by Step

Given $$\int \frac{d x}{\sqrt{16+x^{2}}}$$ Let $$x=4\tan u \ \ \ \ dx=4\sec^2 udu $$ Then \begin{align*} \int \frac{d x}{\sqrt{16+x^{2}}}&=\int \frac{4\sec^2 udu}{\sqrt{16+16\tan^{2}u}}\\ &=\int \frac{4\sec^2 udu}{4\sec u}\\ &=\int \sec udu\\ &=\ln |\sec u+\tan u|+C\\ &= \ln \left|\frac{\sqrt{x^{2}+16}}{4}+\frac{x}{4}\right|+C \end{align*}
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