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: 15


$$\frac{1}{54}\left(\tan ^{-1} \frac{x}{3}+\frac{x}{\sqrt{x^{2}+9}} \cdot \frac{3}{\sqrt{x^{2}+9}}\right)+C$$

Work Step by Step

Given $$\int \frac{d x}{\left(x^{2}+9\right)^{2}}$$ Since \begin{align*} x&=3\tan u\ \ \\ dx&=3\sec^2 udu \end{align*} Then \begin{align*} \int \frac{d x}{\left(x^{2}+9\right)^{2}}&=\int \frac{3\sec^2 udu}{\left(9\tan^2 u+9\right)^{2}}\\ &= \frac{1}{27}\int \frac{du}{\sec^2 u}\\ &=\frac{1}{27}\int\cos^2 udu\\ &= \frac{1}{54} \int (1+\cos 2u)du\\ &= \frac{1}{54} \left(u+ \frac{1}{2}\sin 2u\right)+C\\ &= \frac{1}{54} \left(u+ \sin u\cos u\right)+C\\ &= \frac{1}{54}\left(\tan ^{-1} \frac{x}{3}+\frac{x}{\sqrt{x^{2}+9}} \cdot \frac{3}{\sqrt{x^{2}+9}}\right)+C \end{align*}
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