Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - 8.7 Improper Integrals - Exercises - Page 440: 27


$$\frac{1}{2} $$

Work Step by Step

Given $$\int_{0}^{\infty} \frac{x}{\left(1+x^{2}\right)^{2}} d x $$ Let $ u=1+x^2 \ \ \to \ du=2xdx$ and at $x=0\to u=1$, $ x=\infty\to u=\infty$, then \begin{aligned} \int_{0}^{\infty} \frac{x}{\left(1+x^{2}\right)^{2}} d x &=\int_{1}^{\infty} u^{-2} \cdot \frac{d u}{2} \\ &=\frac{1}{2} \lim _{R \rightarrow \infty}\left[\frac{u^{-1}}{-1}\right]_{1}^{R} \\ &=\frac{1}{2} \lim _{R \rightarrow \infty}\left(1-R^{-1}\right) \\ &=\frac{1}{2} \end{aligned}
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