Calculus: Early Transcendentals 9th Edition

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

Chapter 5 - Section 5.5 - The Substitution Rule - 5.5 Exercises - Page 426: 72

Answer

$0$

Work Step by Step

$$\eqalign{ & \int_{ - \pi /3}^{\pi /3} {{x^4}\sin x} dx \cr & {\text{Let }}f\left( x \right) = {x^4}\sin x{\text{ and evaluate }}f\left( { - x} \right) \cr & f\left( { - x} \right) = {\left( { - x} \right)^4}\sin \left( { - x} \right) \cr & f\left( { - x} \right) = {x^4}\sin \left( { - x} \right) \cr & f\left( { - x} \right) = - {x^4}\sin x \cr & f\left( x \right) = - f\left( x \right),{\text{ therefore }}{x^4}\sin x{\text{ is an odd function}}{\text{, using}} \cr & {\text{the property }}\int_{ - a}^a {f\left( x \right)} dx = 0,{\text{ if }}f\left( x \right){\text{ is odd}}{\text{, we obtain}} \cr & \int_{ - \pi /3}^{\pi /3} {{x^4}\sin x} dx = 0 \cr} $$
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