University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 7 - Section 7.1 - The Logarithm Defined as an Integral - Exercises - Page 402: 45


$$\int\frac{dx}{x\log_{10}x}=\ln10(\ln|\ln x|)+C$$

Work Step by Step

$$A=\int\frac{dx}{x\log_{10}x}=\int\frac{dx}{\frac{x\ln x}{\ln10}}$$ $$A=\ln10\int\frac{dx}{x\ln x}$$ We set $u=\ln x$, which means $$du=\frac{dx}{x}$$ Therefore, $$A=\ln10\int\frac{1}{u}du$$ $$A=\ln10\times\ln|u|+C$$ $$A=\ln10(\ln|\ln x|)+C$$
Update this answer!

You can help us out by revising, improving and updating this answer.

Update this answer

After you claim an answer you’ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide feedback.