Calculus (3rd Edition)

Published by W. H. Freeman
ISBN 10: 1464125260
ISBN 13: 978-1-46412-526-3

Chapter 16 - Multiple Integration - 16.1 Integration in Two Variables - Exercises - Page 847: 35



Work Step by Step

\begin{aligned} \int_{1}^{2} \int_{1}^{3} \frac{\ln (x y) d y d x}{y} &=\int_{1}^{2}\left(\left.\frac{1}{2}[\ln (x y)]^{2}\right|_{1} ^{3 }\right) d x \\ &=\frac{1}{2} \int_{1}^{2}[\ln (3 x)]^{2}-[\ln (x)]^{2} d x \\ &=\frac{1}{2} \int_{1}^{2}[\ln (3 x)]^{2} d x-\frac{1}{2} \int_{1}^{2}[\ln (x)]^{2} d x \\ &=\frac{1}{2}\left[x(\ln 3 x)^{2}-(\ln 3)^{2}\right]-\int_{1}^{2} \ln (3 x) d x-\frac{1}{2}\left[2(\ln 2)^{2}-0\right]+\int_{1}^{2} \ln x d x \\ &=(\ln 6)^{2}-\frac{1}{2}(\ln 3)^{2}-\left[x \ln (3 x)-\left.x\right|_{1} ^{2 }\right]-(\ln 2)^{2}+\left[x \ln x-\left.x\right|_{1} ^{2 }\right]\\ &= (\ln 6)^{2}-\frac{1}{2}(\ln 3)^{2}-(\ln 2)^{2}-2 \ln 6+\ln 3+2 \ln 2 \approx 1.028 \end{aligned}
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