Calculus: Early Transcendentals 8th Edition

Published by Cengage Learning
ISBN 10: 1285741552
ISBN 13: 978-1-28574-155-0

Chapter 5 - Section 5.3 - The Fundamental Theorem of Calculus - 5.3 Exercises - Page 400: 32

Answer

$\displaystyle\int\limits_{\pi/4}^{\pi/3}\csc^{2}\theta d\theta=\dfrac{3-\sqrt{3}}{3}$

Work Step by Step

$\displaystyle\int\limits_{\pi/4}^{\pi/3}\csc^{2}\theta d\theta$ Integrate the expression directly and apply the second part of the fundamental theorem of calculus to get the answer: $\displaystyle\int\limits_{\pi/4}^{\pi/3}\csc^{2}\theta d\theta=\cot\theta\Big|_{\pi/4}^{\pi/3}=\cot\dfrac{\pi}{4}-\cot\dfrac{\pi}{3}=1-\dfrac{\sqrt{3}}{3}=\dfrac{3-\sqrt{3}}{3}$
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