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.

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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:**
$$
f(6) \approx -36.896
$$
$$ f(6) \approx -36.896 $$

The derivative of the function is defined by
. If ,
then use a calculator to find the value of to the
nearest thousandth.

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The derivative of the function is defined by . If , then use a calculator to find the value of to the nearest thousandth.