Calculus: Early Transcendentals 9th Edition

Published by Cengage Learning
ISBN 10: 1337613924
ISBN 13: 978-1-33761-392-7

Chapter 5 - Section 5.4 - Indefinite Integrals and the Net Change Theorem - 5.4 Exercises - Page 416: 50

Answer

$1$

Work Step by Step

$$\eqalign{ &\text{Let }I= \int_{\pi /6}^{\pi /2} {\csc t\cot t} dt \cr & {\text{Integrate using the basic formula }}\int {\csc x\cot x} dx = - \csc x + C \cr &I = \left[ { - \csc t} \right]_{\pi /6}^{\pi /2} \cr & {\text{Using the fundamental theorem of calculus}}{\text{, part 2}} \cr & I=\left[ { - \csc t} \right]_{\pi /6}^{\pi /2} = - \left[ {\csc \left( {\frac{\pi }{2}} \right) - \csc \left( {\frac{\pi }{6}} \right)} \right] \cr & {\text{Simplify}} \cr & I=\left[ { - \csc t} \right]_{\pi /6}^{\pi /2} = - \left[ {1 - 2} \right] = 1 \cr} $$
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