For the function \( f \), given below, find the antiderivative \( F \) that satisfies \( F(1)=2 \). \( f(x)=x^{3}-4 x^{-2}+3 \) The antiderivative that satisfies the given condition is \( F(x)=\square \).

To find the antiderivative \( F(x) \) of the function \( f(x) = x^3 - 4x^{-2} + 3 \) that satisfies \( F(1) = 2 \), follow these steps: 1. **Find the indefinite integral of \( f(x) \):** \[ F(x) = \int f(x) \, dx = \int \left( x^3 - 4x^{-2} + 3 \right) dx \] 2. **Integrate each term separately:** \[ \begin{align*} \int x^3 \, dx &= \frac{x^4}{4} \\ \int -4x^{-2} \, dx &= -4 \cdot \left( -\frac{x^{-1}}{1} \right) = \frac{4}{x} \\ \int 3 \, dx &= 3x \\ \end{align*} \] 3. **Combine the results and add the constant of integration \( C \):** \[ F(x) = \frac{x^4}{4} + \frac{4}{x} + 3x + C \] 4. **Use the initial condition \( F(1) = 2 \) to solve for \( C \):** \[ \frac{1^4}{4} + \frac{4}{1} + 3(1) + C = 2 \\ \frac{1}{4} + 4 + 3 + C = 2 \\ 7.25 + C = 2 \\ C = 2 - 7.25 = -5.25 = -\frac{21}{4} \] 5. **Substitute \( C \) back into the expression for \( F(x) \):** \[ F(x) = \frac{x^4}{4} + \frac{4}{x} + 3x - \frac{21}{4} \] **Final Answer:** \[ F(x) = \frac{x^4}{4} + \frac{4}{\,x\,} + 3\,x\; -\; \frac{21}{4} \]