Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.5 The Method of Partial Fractions - Exercises - Page 423: 36


$$-\frac{1}{25 x}-\frac{1}{125} \tan ^{-1}\left(\frac{x}{5}\right)+C$$

Work Step by Step

Given $$\int \frac{d x}{x^2\left(x^{2}+25\right)}$$ Since \begin{align*} \frac{1}{(x^2 )\left(x^{2}+25\right)}&=\frac{A}{x}+\frac{B}{x^{2}}+\frac{C x+D}{x^{2}+25}\\ &=\frac{A x^{3}+25 A x+B x^{2}+25 B+C x^{3}+D x^{2}}{(x^2 )\left(x^{2}+25\right)} \\ 1&=A x^{3}+25 A x+B x^{2}+25 B+C x^{3}+D x^{2} \end{align*} Then by comparing the coefficients, we get $$A=C=0, \ \ \ B=\frac{1}{25},\ \ \ C=\frac{-1}{25}$$ Hence $B=\frac{-1}{25}$ and \begin{aligned} \int \frac{1}{x^2\left(x^{2}+25\right)} d x &=\frac{1}{25} \int \frac{1}{x^2} d x-\frac{1}{25}\int \frac{ 1}{x^{2}+25} d x \\ &= -\frac{1}{25 x}-\frac{1}{125} \tan ^{-1}\left(\frac{x}{5}\right)+C \end{aligned}
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