Calculus: Early Transcendentals (2nd Edition)

Published by Pearson
ISBN 10: 0321947347
ISBN 13: 978-0-32194-734-5

Chapter 5 - Integration - 5.4 Working with Integrals - 5.4 Exercises: 15

Answer

$$0$$

Work Step by Step

$$\eqalign{ & \int_{ - \pi /4}^{\pi /4} {{{\sin }^5}xdx} \cr & {\text{testing for symmetry}} \cr & f\left( x \right) = {\sin ^5}x \cr & f\left( { - x} \right) = {\left( {\sin \left( { - x} \right)} \right)^5} \cr & f\left( { - x} \right) = {\left( { - \sin \left( x \right)} \right)^5} \cr & f\left( { - x} \right) = - {\sin ^5}x \cr & f\left( { - x} \right) = - f\left( x \right){\text{ so the function }}{\sin ^5}x{\text{ is odd}} \cr & {\text{Use the theorem 5}}{\text{.4}} \cr & {\text{If }}f\left( x \right){\text{ is odd}}{\text{, }}\int_{ - a}^a {f\left( x \right)dx} = 0 \cr & then \cr & \int_{ - \pi /4}^{\pi /4} {{{\sin }^5}xdx} = 0 \cr} $$
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