For the following function \( f \), find the antiderivative \( F \) that satisfies the given condition. \( f(x)=6 x^{3}+5 \sin x, F(0)=3 \) The antiderivative that satisfies the given condition is \( F(x)=\square \).

To find the antiderivative \( F(x) \) of the function \( f(x) = 6x^3 + 5\sin x \) that satisfies the condition \( F(0) = 3 \), follow these steps: 1. **Find the indefinite integral of \( f(x) \):** \[ F(x) = \int f(x) \, dx = \int (6x^3 + 5\sin x) \, dx \] Compute each term separately: \[ \int 6x^3 \, dx = \frac{6}{4}x^4 = \frac{3}{2}x^4 \] \[ \int 5\sin x \, dx = -5\cos x \] Combine the results: \[ F(x) = \frac{3}{2}x^4 - 5\cos x + C \] 2. **Apply the initial condition \( F(0) = 3 \) to find the constant \( C \):** \[ F(0) = \frac{3}{2}(0)^4 - 5\cos(0) + C = -5(1) + C = -5 + C = 3 \] Solve for \( C \): \[ C = 3 + 5 = 8 \] 3. **Write the final expression for \( F(x) \):** \[ F(x) = \frac{3}{2}x^4 - 5\cos x + 8 \] **Answer:** \( F(x) = \dfrac{3}{2}\, x^{4} - 5\,\cos x + 8 \)