## Calculus with Applications (10th Edition)

$$y = {x^3} + 3{x^2} + C$$
\eqalign{ & \frac{{dy}}{{dx}} = 3{x^2} + 6x \cr & {\text{Separating variables leads to}} \cr & dy = \left( {3{x^2} + 6x} \right)dx \cr & {\text{To solve this equation}}{\text{, determine the antiderivative of each side}} \cr & \int {dy} = \int {\left( {3{x^2} + 6x} \right)dx} \cr & {\text{integrating by using the power rule we obtain}} \cr & y = 3\left( {\frac{{{x^3}}}{3}} \right) + 6\left( {\frac{{{x^2}}}{2}} \right) + C \cr & {\text{simplifying}} \cr & y = {x^3} + 3{x^2} + C \cr}