Thomas' Calculus 13th Edition

Published by Pearson
ISBN 10: 0-32187-896-5
ISBN 13: 978-0-32187-896-0

Chapter 7: Transcendental Functions - Practice Exercises - Page 439: 44


$$\ln \left| {\ln v} \right| + C $$

Work Step by Step

$$\eqalign{ & \int {\frac{{dv}}{{v\ln v}}} \cr & {\text{Integrate by the substitution method}} \cr & {\text{set }}u = \ln v{\text{ then }}\frac{{du}}{{dv}} = \frac{1}{v},\,\,\,\,dv = vdu \cr & {\text{write the integrand in terms of }}u \cr & \int {\frac{{dv}}{{v\ln v}}} = \int {\frac{1}{{vu}}} \left( {vdu} \right) \cr & {\text{cancel common terms}} \cr & = \int {\frac{1}{u}} du \cr & {\text{integrating}}{\text{,}} \cr & = \ln \left| u \right| + C \cr & {\text{replace }}\ln v{\text{ for }}u \cr & = \ln \left| {\ln v} \right| + C \cr} $$
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