Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 460: 44


$$\frac{1}{250} \tan ^{-1}\left(\frac{x}{5}\right)+\frac{x}{50\left(x^{2}+25\right)}+C$$

Work Step by Step

Given $$\int \frac{d x}{\left(x^{2}+25\right)^{2}} $$ Let $$x=5\tan u\ \ \ \ \ \ \ \ \ \ \ \ dx=5\sec^2 udu $$ Then \begin{align*} \int \frac{d x}{\left(x^{2}+25\right)^{2}}&=\int \frac{5\sec^2 udu}{\left(25\tan^{2}u+25\right)^{2}}\\ &=\int \frac{5\sec^2 udu}{\left(25\sec^{2}u\right)^{2}}\\ &= \frac{1}{125}\int \frac{du}{\sec^2 u}\\ &= \frac{1}{125}\int \cos^2 udu\\ &= \frac{1}{250}\int (1+\cos2 u)du\\ &=\frac{1}{250}(u+\frac{1}{2}\sin 2u)+C\\ &=\frac{1}{250}(u+ \sin u\cos u)+C\\ &=\frac{1}{250} \tan ^{-1} \frac{x}{5}+\frac{1}{250} \cdot \frac{5}{\sqrt{x^{2}+25}} \cdot \frac{x}{\sqrt{x^{2}+25}}+C\\ &=\frac{1}{250} \tan ^{-1}\left(\frac{x}{5}\right)+\frac{x}{50\left(x^{2}+25\right)}+C \end{align*}
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