Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 6 - Applications of Integration - Review Exercises: 60

Answer

$$\ln \left( {\frac{{16}}{9}} \right)$$

Work Step by Step

$$\eqalign{ & \int_{\ln 2}^{\ln 3} {\coth s} ds \cr & {\text{hyperbolic function}} \cr & = \int_{\ln 2}^{\ln 3} {\frac{{\cosh s}}{{\sinh s}}} ds \cr & {\text{substitute }}u = \sinh s,{\text{ }}du = \cosh sds \cr & {\text{express the limits in terms of }}u \cr & x = \ln 3{\text{ implies }}u = \sinh s = 4/3 \cr & x = \ln 2{\text{ implies }}u = \sinh s = 3/4 \cr & {\text{the entire integration is carried out as follows}} \cr & \int_{\ln 2}^{\ln 3} {\frac{{\cosh s}}{{\sinh s}}} ds = \int_{3/4}^{4/3} {\frac{{du}}{u}} \cr & {\text{find the antiderivative}} \cr & = \left. {\left( {\ln \left| u \right|} \right)} \right|_{3/4}^{4/3} \cr & = \ln \left( {\frac{4}{3}} \right) - \ln \left( {\frac{3}{4}} \right) \cr & = \ln \left( {\frac{{16}}{9}} \right) \cr} $$
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