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: 104

Answer

$$\sin 4w + \cos 3w + C$$

Work Step by Step

$$\eqalign{ & \int {\left( {4\cos 4w - 3\sin 3w} \right)dw} \cr & {\text{split the integrand}} \cr & = \int {4\cos 4wdw} - \int {3\sin 3wdw} \cr & = 4\int {\cos 4wdw} - 3\int {\sin 3wdw} \cr & {\text{use the formula for indefinite integrals of trigonometric functions}} \cr & = 4\left( {\frac{1}{4}\sin 4w} \right) - 3\left( { - \frac{1}{3}\cos 3w} \right) + C \cr & {\text{simplify}} \cr & = \sin 4w + \cos 3w + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{dw}}\left( {\sin 4w + \cos 3w + C} \right) \cr & {\text{ = }}\frac{d}{{dw}}\left( {\sin 4w} \right) + \frac{d}{{dw}}\left( {\cos 3w + C} \right) + {\text{ = }}\frac{d}{{dw}}\left( C \right) \cr & {\text{ = }}4\cos 4w - 3\sin 3w + 0 \cr & = 4\cos 4w - 3\sin 3w \cr} $$
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