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: 41

Answer

$$\tan \theta + \sec \theta + C$$

Work Step by Step

$$\eqalign{ & \int {\left( {{{\sec }^2}\theta + \sec \theta \tan \theta } \right)} d\theta \cr & {\text{sum rule}} \cr & \int {{{\sec }^2}\theta } d\theta + \int {\sec \theta \tan \theta } d\theta \cr & {\text{use the formula for indefinite integrals of trigonometric functions}} \cr & \tan \theta + \sec \theta + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{d\theta }}\left( {\tan \theta + \sec \theta + C} \right) \cr & {\text{ = }}\frac{d}{{d\theta }}\left( {\tan \theta } \right) + \frac{d}{{d\theta }}\left( {\sec \theta } \right) + \frac{d}{{d\theta }}\left( C \right) \cr & = {\sec ^2}\theta + \sec \theta \tan \theta + 0 \cr & = {\sec ^2}\theta + \sec \theta \tan \theta \cr} $$
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