Calculus: Early Transcendentals (2nd Edition)

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

Chapter 4 - Applications of the Derivative - 4.9 Antiderivatives - 4.9 Exercises: 37

Answer

$$ - \frac{1}{2}\cos 2y + \frac{1}{3}\sin 3y + C$$

Work Step by Step

$$\eqalign{ & \int {\left( {\sin 2y + \cos 3y} \right)} dy \cr & {\text{sum Rule}} \cr & = \int {\sin 2y} dy + \int {\cos 3y} dy \cr & {\text{use the formula for indefinite integrals of trigonometric functions}} \cr & \int {\sin ax} dx = - \frac{1}{a}\cos ax + C,{\text{ }}\int {\cos axdx} = \frac{1}{a}sinax + C \cr & = - \frac{1}{2}\cos 2y + \frac{1}{3}\sin 3y + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{dx}}\left( { - \frac{1}{2}\cos 2y + \frac{1}{3}\sin 3y + C} \right) \cr & = - \frac{1}{2}\left( { - 2\sin 2y} \right) + \frac{1}{3}\left( {3\cos 3y} \right) + 0 \cr & = \sin 2y + \cos 3y \cr} $$
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