1. **State the problem:** We need to find the integral of the function $y = 3x^2 + 2x - 5$ with respect to $x$. 2. **Recall the formula:** The integral of a polynomial term $ax^n$ with respect to $x$ is given by $$\int ax^n \, dx = \frac{a}{n+1}x^{n+1} + C$$ where $C$ is the constant of integration. 3. **Apply the integral to each term:** $$\int (3x^2 + 2x - 5) \, dx = \int 3x^2 \, dx + \int 2x \, dx - \int 5 \, dx$$ 4. **Integrate each term:** - For $3x^2$: $$\int 3x^2 \, dx = 3 \cdot \frac{x^{2+1}}{2+1} = 3 \cdot \frac{x^3}{3} = \cancel{3} \cdot \frac{x^3}{\cancel{3}} = x^3$$ - For $2x$: $$\int 2x \, dx = 2 \cdot \frac{x^{1+1}}{1+1} = 2 \cdot \frac{x^2}{2} = \cancel{2} \cdot \frac{x^2}{\cancel{2}} = x^2$$ - For $-5$: $$\int -5 \, dx = -5x$$ 5. **Combine the results and add the constant of integration:** $$\int (3x^2 + 2x - 5) \, dx = x^3 + x^2 - 5x + C$$ **Final answer:** $$\boxed{x^3 + x^2 - 5x + C}$$