Calculus 10th Edition

Published by Brooks Cole
ISBN 10: 1-28505-709-0
ISBN 13: 978-1-28505-709-5

Chapter 5 - Logarithmic, Exponential, and Other Transcendental Functions - 5.2 Exercises - Page 334: 22


$\displaystyle \frac{1}{3}\ln|\ln x|+C$

Work Step by Step

First, $\ln x^{3}=3\ln x$ , so we rewrite the integral $...=\displaystyle \frac{1}{3}\int\frac{1}{\ln x}\cdot\frac{1}{x}dx, \quad\left[\begin{array}{l} u=\ln x\\ du=\frac{1}{x}dx \end{array}\right]$ $=\displaystyle \frac{1}{3}\int\frac{1}{u}du$= ... use Th.5.5.2 $=\displaystyle \frac{1}{3}\ln|u|+C$ $=\displaystyle \frac{1}{3}\ln|\ln x|+C$
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