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: 65

Answer

$${y^3} + 5\ln \left| y \right| + 2$$

Work Step by Step

$$\eqalign{ & f\left( y \right) = \frac{{3{y^3} + 5}}{y} \cr & f\left( y \right) = \frac{{3{y^3}}}{y} + \frac{5}{y} \cr & f\left( y \right) = 3{y^2} + \frac{5}{y} \cr & {\text{find an antiderivative of }}f\left( y \right) \cr & F\left( y \right) = {y^3} + 5\ln \left| y \right| + C \cr & {\text{using the initial condition }}F\left( 1 \right) = 3 \cr & 3 = {\left( 1 \right)^3} + 5\ln \left| 1 \right| + C \cr & 3 = 1 + 5\left( 0 \right) + C \cr & 3 = 1 + 0 + C \cr & C = 2 \cr & = {y^3} + 5\ln \left| y \right| + 2 \cr} $$
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