1. The problem is to find the indefinite integral of the polynomial function $3x^2 + 2x - 1$ with respect to $x$. 2. The formula for integrating a power function $x^n$ is: $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $C$ is the constant of integration. 3. Apply the integral to each term separately: $$\int (3x^2 + 2x - 1) \, dx = \int 3x^2 \, dx + \int 2x \, dx - \int 1 \, dx$$ 4. Integrate each term: - For $3x^2$: $$3 \int x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3}$$ - For $2x$: $$2 \int x \, dx = 2 \cdot \frac{x^{1+1}}{1+1} = 2 \cdot \frac{x^2}{2}$$ - For $-1$: $$- \int 1 \, dx = -x$$ 5. Simplify each term: $$3 \cdot \frac{x^3}{3} = \cancel{3} \cdot \frac{x^3}{\cancel{3}} = x^3$$ $$2 \cdot \frac{x^2}{2} = \cancel{2} \cdot \frac{x^2}{\cancel{2}} = x^2$$ 6. Combine all terms and add the constant of integration $C$: $$x^3 + x^2 - x + C$$ Therefore, the integral is: $$\int (3x^2 + 2x - 1) \, dx = x^3 + x^2 - x + C$$