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: 35


$$\frac{1}{25} \ln |x|-\frac{1}{50} \ln \left|x^{2}+25\right|+C$$

Work Step by Step

Given $$\int \frac{d x}{x\left(x^{2}+25\right)}$$ Since \begin{align*} \frac{1}{(x )\left(x^{2}+25\right)}&=\frac{A}{x }+\frac{Bx+C}{x^{2}+25}\\ &=\frac{A(x^{2}+25)+B(x)}{(x)\left(x^{2}+25\right)}\\ 1&=A(x^{2}+25)+B(x) \end{align*} \begin{align*} \text{at } x&=0 \ \ \ \ \to A=\frac{1}{25} \\ \text{at } x&=1\ \ \ \ \to B+C=\frac{-1}{25}\\ \text{at } x&= -1\ \ \ \ \to B -C=\frac{-1}{25} \end{align*} Hence $B=\frac{-1}{25}$ and \begin{aligned} \int \frac{1}{x\left(x^{2}+25\right)} d x &=\int \frac{\frac{1}{25}}{x} d x-\int \frac{ \frac{1}{25} x}{x^{2}+25} d x \\ &=\frac{1}{25} \ln |x|-\frac{1}{50} \ln \left|x^{2}+25\right|+C \end{aligned}
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