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 \( f(x) = 6x^3 + 5 \sin x \), we first integrate the function: \[ F(x) = \int (6x^3 + 5 \sin x) \, dx = \int 6x^3 \, dx + \int 5 \sin x \, dx. \] Calculating each integral separately, we get: \[ \int 6x^3 \, dx = \frac{6}{4} x^4 = \frac{3}{2} x^4, \] and \[ \int 5 \sin x \, dx = -5 \cos x. \] Thus, \[ F(x) = \frac{3}{2} x^4 - 5 \cos x + C, \] where \( C \) is the constant of integration. Next, we apply the initial condition \( F(0) = 3 \): \[ F(0) = \frac{3}{2} (0)^4 - 5 \cos(0) + C = 0 - 5 + C = C - 5. \] Setting this equal to 3 gives us: \[ C - 5 = 3 \implies C = 3 + 5 = 8. \] Finally, we substitute \( C \) back into our expression for \( F(x) \): \[ F(x) = \frac{3}{2} x^4 - 5 \cos x + 8. \] So, the antiderivative that satisfies the given condition is \[ F(x) = \frac{3}{2} x^4 - 5 \cos x + 8. \]