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: 17



Work Step by Step

\begin{aligned} \int_{-1}^{1} \frac{d x}{x^{0.2}} &=\int_{-1}^{0} \frac{d x}{x^{0.2}}+\int_{0}^{1} \frac{d x}{x^{0.2}} \\ &=\lim _{R \rightarrow 0^{-}} \int_{-1}^{R} \frac{d x}{x^{0.2}}+\lim _{R \rightarrow 0^{+}} \int_{R}^{1} \frac{d x}{x^{0.2}} \\ &=\left.\lim _{R \rightarrow 0^{-}} \frac{x^{0.8}}{0.8}\right|_{-1} ^{R}+\left.\lim _{R \rightarrow 0^{+}} \frac{x^{0.8}}{0.8}\right|_{R} ^{1} \\ &=\lim _{R \rightarrow 0^{-}} \frac{R^{0.8}-1}{0.8}+\lim _{R \rightarrow 0^{+}} \frac{1-R^{0.8}}{0.8} \\ &=-\frac{5}{4}+\frac{5}{4}\\ &=0 \end{aligned}
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