Calculus: Early Transcendentals (2nd Edition)

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

Chapter 7 - Integration Techniques - 7.8 Improper Integrals - 7.8 Exercises - Page 578: 18



Work Step by Step

$$\eqalign{ & \int_0^\infty {\cos xdx} \cr & {\text{definition of improper integral}} \cr & \int_0^\infty {\cos xdx} = \mathop {\lim }\limits_{b \to \infty } \int_0^b {\cos xdx} \cr & {\text{evaluate the integral}} \cr & = \mathop {\lim }\limits_{b \to \infty } \left. {\left( {\sin x} \right)} \right|_0^b \cr & = \mathop {\lim }\limits_{b \to \infty } \left. {\left( {\sin b - \sin 0} \right)} \right| \cr & = \mathop {\lim }\limits_{b \to \infty } \left. {\left( {\sin b} \right)} \right| \cr & {\text{evaluate the limit}} \cr & = \sin \left( \infty \right) \cr & {\text{the limit does not exist}}{\text{, the integral diverges}} \cr & {\text{diverges}} \cr} $$
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