1. **Problem Statement:** Calculate the integral $\int (3x^2 + 2x + 1) \, dx$. 2. **Formula Used:** The integral of a sum is the sum of the integrals, and the power rule for integration states: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $n \neq -1$. 3. **Step-by-step Solution:** - Break the integral into parts: $$\int (3x^2 + 2x + 1) \, dx = \int 3x^2 \, dx + \int 2x \, dx + \int 1 \, dx$$ - Integrate each term: $$\int 3x^2 \, dx = 3 \int x^2 \, dx = 3 \cdot \frac{x^{3}}{3} = \cancel{3} \cdot \frac{x^{3}}{\cancel{3}} = x^{3}$$ $$\int 2x \, dx = 2 \int x \, dx = 2 \cdot \frac{x^{2}}{2} = \cancel{2} \cdot \frac{x^{2}}{\cancel{2}} = x^{2}$$ $$\int 1 \, dx = x$$ 4. **Combine all results:** $$x^{3} + x^{2} + x + C$$ 5. **Final Answer:** $$\int (3x^2 + 2x + 1) \, dx = x^{3} + x^{2} + x + C$$ This means the antiderivative of the polynomial $3x^2 + 2x + 1$ is $x^{3} + x^{2} + x + C$, where $C$ is the constant of integration.