Calculus (3rd Edition)

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

Chapter 7 - Exponential Functions - Chapter Review Exercises - Page 387: 86


$$\ln 3.$$

Work Step by Step

We have $$\int_{\pi/3}^{2\pi/3}\cot(\theta/2) d\theta=\int_{\pi/3}^{2\pi/3}\frac{\cos(\theta/2)}{\sin(\theta/2)} d\theta\\ =2\int_{\pi/3}^{2\pi/3}\frac{(\sin(\theta/2))'}{\sin(\theta/2)}=2\ln\sin(\theta/2)|_{\pi/3}^{2\pi/3}\\ =2(\ln\sin(\pi/3)-\ln \sin (\pi/6))=2\ln(\sqrt 3/2)-2\ln (1/2)=2\ln\sqrt 3=\ln 3.$$
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