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