1. **State the problem:** We need to find the indefinite integral of the polynomial function $$8x^2 - 3x + 7$$ with respect to $$x$$. 2. **Recall the formula for integration of power functions:** $$\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$$ where $$n \neq -1$$ and $$C$$ is the constant of integration. 3. **Apply the integral to each term separately:** $$\int (8x^2 - 3x + 7) \, dx = \int 8x^2 \, dx - \int 3x \, dx + \int 7 \, dx$$ 4. **Integrate each term:** - For $$8x^2$$: $$8 \int x^2 \, dx = 8 \cdot \frac{x^{2+1}}{2+1} = 8 \cdot \frac{x^3}{3} = \frac{8}{3}x^3$$ - For $$-3x$$: $$-3 \int x \, dx = -3 \cdot \frac{x^{1+1}}{1+1} = -3 \cdot \frac{x^2}{2} = -\frac{3}{2}x^2$$ - For constant $$7$$: $$7 \int 1 \, dx = 7x$$ 5. **Combine all integrated terms and add the constant of integration $$C$$:** $$\int (8x^2 - 3x + 7) \, dx = \frac{8}{3}x^3 - \frac{3}{2}x^2 + 7x + C$$ This is the final answer.