Calculus 8th Edition

Published by Cengage
ISBN 10: 1285740629
ISBN 13: 978-1-28574-062-1

Chapter 7 - Techniques of Integration - 7.2 Trigonometric Integrals - 7.2 Exercises - Page 525: 35



Work Step by Step

$1+cot^{2}x=csc^{2}x,\,\,\int csc^{2}x\,dx=-cotx+C$ So$$\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}cot^{2}x\,dx=\int_{\frac{\pi}{6}}^{\frac{\pi}{2}}(csc^{2}x-1)dx$$ $$=\left |-cotx-x \right |_{\frac{\pi}{6}}^{\frac{\pi}{2}}$$ $$=0-\frac{\pi}{2}+\sqrt{3}+\frac{\pi}{6}$$ $$=\sqrt{3}-\frac{\pi}{3}$$
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