General antiderivative of a polynomial

Apply the power rule in reverse to each term separately: $\int 4x^3\,dx = 4 \cdot \dfrac{x^4}{4} = x^4$, $\int -6x^2\,dx = -6 \cdot \dfrac{x^3}{3} = -2x^3$, $\int 2x\,dx = 2 \cdot \dfrac{x^2}{2} = x^2$, and $\int -5\,dx = -5x$.

Adding these together and including a single arbitrary constant $C$ to represent all possible antiderivatives gives $F(x) = x^4 - 2x^3 + x^2 - 5x + C$. This $C$ is essential since any constant added to $F$ still has derivative $f(x)$.