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: 45

Answer

$$\frac{1}{4}sec4\theta + C$$

Work Step by Step

$$\eqalign{ & \int {\sec 4\theta tan4\theta } d\theta \cr & {\text{use the formula for indefinite integrals of trigonometric functions}} \cr & \int {\sec ax} \tan axdx = \frac{1}{a}secax + C \cr & {\text{letting }}a = 4,{\text{ and }}x = \theta ,{\text{ we have}} \cr & = \frac{1}{4}sec4\theta + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{d\theta }}\left( {\frac{1}{4}sec4\theta + C} \right) \cr & {\text{ = }}\frac{d}{{d\theta }}\left( {\frac{1}{4}sec4\theta } \right) + \frac{d}{{d\theta }}\left( C \right) \cr & = \frac{1}{4}\left( {\sec 4\theta tan4\theta } \right)\left( 4 \right) + 0 \cr & = \sec 4\theta tan4\theta \cr} $$
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