The derivative of the function \( f \) is defined by \( f^{\prime}(x)=\left(x^{2}+4 x\right) \cos (2 x+5) \). If \( f(-2)=-8 \), then use a calculator to find the value of \( f(6) \) to the nearest thousandth.

To find \( f(6) \) given the derivative \( f^{\prime}(x) = (x^2 + 4x) \cos(2x + 5) \) and the initial condition \( f(-2) = -8 \), we'll perform the following steps: 1. **Integration of \( f^{\prime}(x) \):** We need to compute the integral: \[ \int_{-2}^{6} (x^2 + 4x) \cos(2x + 5) \, dx \] Using integration by parts, we find the antiderivative \( F(x) \): \[ F(x) = \left(\frac{1}{2}x^2 + 2x - \frac{1}{4}\right) \sin(2x + 5) + (x + 2) \cos(2x + 5) \] 2. **Evaluating the Integral from \(-2\) to \(6\):** \[ F(6) - F(-2) = \left[\frac{1}{2}(6)^2 + 2(6) - \frac{1}{4}\right] \sin(17) + (6 + 2) \cos(17) - \left[\frac{1}{2}(-2)^2 + 2(-2) - \frac{1}{4}\right] \sin(1) + (-2 + 2) \cos(1) \] Simplifying further: \[ F(6) - F(-2) \approx 29.75 \cdot (-0.9613) + 8 \cdot (-0.2756) + 2.25 \cdot 0.8415 \approx -28.896 \] 3. **Adding the Initial Condition:** Given \( f(-2) = -8 \): \[ f(6) = f(-2) + (F(6) - F(-2)) \approx -8 + (-28.896) = -36.896 \] **Final Answer:** \[ f(6) \approx -36.896 \]