Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 7 - Techniques of Integration - Review - Exercises - Page 578: 43

Answer

Divergent

Work Step by Step

Let \[I=\int_{2}^{\infty}\frac{dx}{x\ln x}\] This is improper integral \[\Rightarrow I=\lim_{t\rightarrow \infty}\int_{2}^{t}\frac{dx}{x\ln x} \;\;\;\ldots (1)\] \[I=\lim_{t\rightarrow \infty}\left[\ln|\ln x|\right]_{2}^{t}\] \[I=\lim_{t\rightarrow \infty}\left[\ln|\ln t|-\ln|\ln 2|\right]\] $\;\;\;\;\;\;\;\;\;\;\;\;\;\Rightarrow$does not exist Since limit on R.H.S. of (1) does not exist so $I$ is divergent.
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