University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 8 - Section 8.7 - Improper Integrals - Exercises - Page 471: 22



Work Step by Step

Here, we have $\lim\limits_{a \to \infty}\int_0^{a} e^{-\theta} \sin \theta d\theta=\lim\limits_{a \to \infty}[ -e^{-\theta} (\cos \theta+\sin \theta)|_0^{a}$ This implies that $\lim\limits_{a \to \infty}[ -e^{-\theta} (\cos \theta+\sin \theta)|_0^{a}=\lim\limits_{a \to \infty}[ -e^{-a} (\cos a+\sin a-(-e^{-0} (\cos 0+\sin 0)]$ or, $\lim\limits_{a \to \infty}[ -e^{-a} (\cos a+\sin a-(-e^{-0} (\cos 0+\sin 0)]=\lim\limits_{a \to \infty}[ -e^{-a} (\cos a+\sin a)+1]$ Thus, $\lim\limits_{a \to \infty}[ -e^{-a} (\cos a+\sin a)+1]=0+1=1$
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