1. **Problem:** Find the indefinite integral (antiderivative) of the function $4x^3 - 6x + 9$. 2. **Formula:** The integral of a sum is the sum of the integrals, and the power rule for integration states: $$\int x^n dx = \frac{x^{n+1}}{n+1} + C, \quad n \neq -1$$ 3. **Step-by-step solution:** - Integrate each term separately: $$\int 4x^3 dx = 4 \cdot \frac{x^{4}}{4} = x^4$$ $$\int (-6x) dx = -6 \cdot \frac{x^{2}}{2} = -3x^2$$ $$\int 9 dx = 9x$$ 4. **Combine results:** $$\int (4x^3 - 6x + 9) dx = x^4 - 3x^2 + 9x + C$$ 5. **Explanation:** The constant $C$ represents the family of antiderivatives since differentiation of a constant is zero. **Final answer:** $$\boxed{x^4 - 3x^2 + 9x + C}$$