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: 57

Answer

$${e^{x + 2}} + C$$

Work Step by Step

$$\eqalign{ & \int {{e^{x + 2}}} dx \cr & {\text{property }}{e^{a + b}} = {e^a}{e^b} \cr & = \int {{e^2}{e^x}} dx \cr & = {e^2}\int {{e^x}} dx \cr & {\text{integrate}} \cr & = {e^2}\left( {{e^x}} \right) + C \cr & = {e^{x + 2}} + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{dx}}\left( {{e^{x + 2}} + C} \right) \cr & {\text{ = }}\frac{d}{{dx}}\left( {{e^{x + 2}}} \right) + \frac{d}{{dx}}\left( C \right) \cr & {\text{ = }}{e^2}\frac{d}{{dx}}\left( {{e^x}} \right) + \frac{d}{{dx}}\left( C \right) \cr & {\text{ = }}{e^2}\left( {{e^x}} \right) + 0 \cr & = {e^{x + 2}} \cr} $$
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