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: 32

Answer

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

Work Step by Step

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