Calculus (3rd Edition)

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

Chapter 8 - Techniques of Integration - Chapter Review Exercises - Page 461: 86



Work Step by Step

Given $$ \int_{8}^{\infty}\left(\sin ^{2} x\right) e^{-x} d x$$ Since for $x>0$ $$0 \leq\left(\sin ^{2} x\right) e^{-x} \leq e^{-x}$$ and \begin{aligned} \int_{8}^{\infty} e^{-x} d x &=\lim _{R \rightarrow \infty} \int_{8}^{R} e^{-x} d x \\ &=\lim _{R \rightarrow \infty}-\left.e^{-x}\right|_{8} ^{R} \\ &=\lim _{R \rightarrow \infty}\left(-e^{-R}+e^{-8}\right) \\ &=e^{-8} \end{aligned} Then by the comparison test, $ \int_{8}^{\infty}\left(\sin ^{2} x\right) e^{-x} d x$ also converges
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