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: 55

Answer

$$t + \ln \left| t \right| + C$$

Work Step by Step

$$\eqalign{ & \int {\frac{{t + 1}}{t}dt} \cr & = \int {\left( {\frac{t}{t} + \frac{1}{t}} \right)dt} \cr & = \int {\left( {1 + \frac{1}{t}} \right)dt} \cr & {\text{sum rule}} \cr & = \int {dt} + \int {\frac{1}{t}dt} \cr & {\text{integrate}} \cr & = t + \ln \left| t \right| + C \cr & {\text{check by differentiation}} \cr & {\text{ = }}\frac{d}{{dt}}\left( {t + \ln \left| t \right| + C} \right) \cr & {\text{ = }}\frac{d}{{dt}}\left( t \right) + \frac{d}{{dt}}\left( {\ln \left| t \right|} \right) + \frac{d}{{dt}}\left( C \right) \cr & {\text{ = }}1 + \frac{1}{t} + 0 \cr & add \cr & = \frac{{t + 1}}{t} \cr} $$
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