Answer
The integral diverges.
Work Step by Step
We have
$$\int_1^2\frac{dx}{x \ln x}=\int_0^{\ln 2}\frac{d(\ln x)}{\ln x}\\
=\ln(\ln x)|_0^{\ln 2}=\ln (\ln 2)-\ln 0=-\infty
$$
Hence, the integral diverges.
Calculus (3rd Edition)
by Rogawski, Jon; Adams, Colin
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 441: 38
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