Thomas' Calculus 13th Edition

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

Chapter 8: Techniques of Integration - Practice Exercises - Page 519: 108


$$\ln \left| {\ln \left( {\ln x} \right)} \right| + C$$

Work Step by Step

$$\eqalign{ & \int {\frac{1}{{x \cdot \ln x \cdot \ln \left( {\ln x} \right)}}} dx \cr & {\text{Write the integrand as}} \cr & \int {\left( {\frac{1}{{\ln \left( {\ln x} \right)}}} \right)\left( {\frac{1}{{x\ln x}}} \right)} dx \cr & {\text{Integrate by using the substitution method}}{\text{,}} \cr & {\text{Let }}u = \ln \left( {\ln x} \right),\,\,\,\,\,\,\,\,\,du = \frac{{\frac{1}{x}}}{{\ln x}}dx,\,\,\,\,\,\,du = \frac{1}{{x\ln x}}dx \cr & \cr & {\text{Write the integrand in terms of }}u \cr & \,\,\int {\left( {\frac{1}{{\ln \left( {\ln x} \right)}}} \right)\left( {\frac{1}{{x\ln x}}} \right)} dx = \int {\frac{1}{u}du} \cr & {\text{Integrate}} \cr & = \ln \left| u \right| + C \cr & \cr & {\text{Write in terms of }}x,{\text{ substitute }}\ln \left( {\ln x} \right){\text{ for }}u \cr & = \ln \left| {\ln \left( {\ln x} \right)} \right| + C \cr} $$
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