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: 12

Answer

Diverges

Work Step by Step

\begin{aligned} \int_{5}^{6} \frac{d x}{(x-5)^{3 / 2}} &=\lim _{R \rightarrow 5} \int_{R}^{6} \frac{d x}{(x-5)^{3 / 2}} \\ &=\lim _{R \rightarrow 5}\left.\frac{(x-5)^{(-1 / 2)}}{-1 / 2}\right|_{R} ^{6} \\ &=\lim _{R \rightarrow 5} \left(\frac{2}{(R-5)^{(1 / 2)}}-2 \right)\\ &=\infty \end{aligned} Then integral diverges.
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