1. The problem is to find the function $y$ given its derivative $\frac{dy}{dx} = x^3 + 2x$. 2. We use the formula for antiderivatives: if $\frac{dy}{dx} = f(x)$, then $y = \int f(x) \, dx + C$ where $C$ is the constant of integration. 3. Integrate each term separately: $$y = \int x^3 \, dx + \int 2x \, dx + C$$ 4. Using the power rule for integration $\int x^n \, dx = \frac{x^{n+1}}{n+1}$ for $n \neq -1$: $$y = \frac{x^{4}}{4} + 2 \cdot \frac{x^{2}}{2} + C$$ 5. Simplify the expression: $$y = \frac{x^{4}}{4} + x^{2} + C$$ 6. This is the general solution for $y$ given the derivative. Final answer: $$y = \frac{x^{4}}{4} + x^{2} + C$$