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: 48

Answer

$$\frac{1}{2}{e^{2t}} + \frac{4}{3}{t^{3/2}} + C$$

Work Step by Step

$$\eqalign{ & \int {\left( {{e^{2t}} + 2\sqrt t } \right)} dt \cr & {\text{use radical property}} \cr & = \int {\left( {{e^{2t}} + 2{t^{1/2}}} \right)} dt \cr & {\text{split the integrand}} \cr & = \int {{e^{2t}}} dt + \int {2{t^{1/2}}} dt \cr & {\text{integrate}} \cr & = \frac{1}{2}\left( {{e^{2t}}} \right) + 2\left( {\frac{{{t^{3/2}}}}{{3/2}}} \right) + C \cr & = \frac{1}{2}{e^{2t}} + \frac{4}{3}{t^{3/2}} + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{dt}}\left( {\frac{1}{2}{e^{2t}} + \frac{4}{3}{t^{3/2}} + C} \right) \cr & {\text{ = }}\frac{d}{{dt}}\left( {\frac{1}{2}{e^{2t}}} \right) + \frac{d}{{dt}}\left( {\frac{4}{3}{t^{3/2}}} \right) + \frac{d}{{dt}}\left( C \right) \cr & {\text{ = }}\frac{1}{2}\left( {{e^{2t}}} \right)\left( 2 \right) + \frac{4}{3}\left( {\frac{3}{2}} \right){t^{1/2}} + 0 \cr & = {e^{2t}} + 2\sqrt t \cr} $$
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