To approximate the integral $\int_0^\pi \ln(6 + 5\cos(x)) \, dx$ using Simpson's Rule, we need to compute the integral for different values of $n$ and compare the results to ensure the error is less than $10^{-3}$. Simpson's Rule is given by:

$\text{Simpson's Rule} \approx \frac{\Delta x}{3} \left[ f(x_0) + 4\sum_{i=1,3,5,\ldots}^{n-1} f(x_i) + 2\sum_{i=2,4,6,\ldots}^{n-2} f(x_i) + f(x_n) \right]$

Where:

• $\Delta x = \frac{b-a}{n}$ is the width of each subinterval.
• $f(x_i)$ are the function values at the endpoints and interior points of the intervals.

For this problem, $a = 0$, $b = \pi$, and $f(x) = \ln(6 + 5\cos(x))$.

Let's compute the approximation for different values of $n$.

For $n = 4$:

• Compute $\Delta x = \frac{\pi - 0}{4} = \frac{\pi}{4}$.
• The points are $x_0 = 0$, $x_1 = \frac{\pi}{4}$, $x_2 = \frac{\pi}{2}$, $x_3 = \frac{3\pi}{4}$, and $x_4 = \pi$.
• Compute the function values at these points.

For $n = 8$:

• Compute $\Delta x = \frac{\pi - 0}{8} = \frac{\pi}{8}$.
• The points are $x_0 = 0$, $x_1 = \frac{\pi}{8}$, $x_2 = \frac{\pi}{4}$, and so on up to $x_8 = \pi$.
• Compute the function values at these points.

For $n = 16$:

• Compute $\Delta x = \frac{\pi - 0}{16} = \frac{\pi}{16}$.
• The points are $x_0 = 0$, $x_1 = \frac{\pi}{16}$, $x_2 = \frac{\pi}{8}$, and so on up to $x_{16} = \pi$.
• Compute the function values at these points.

After calculating the values, check if the approximation converges such that the difference between successive approximations is less than $10^{-3}$.

Since you mentioned that for $n = 32$, the approximation is $4.8338$, we can now calculate the approximations for $n = 4$, $n = 8$, and $n = 16$.

Would you like me to perform these calculations and provide the approximations for the missing values?

---

Relative Questions:

1. How does the error in Simpson's Rule approximation change as $n$ increases?
2. What is the significance of using Simpson's Rule instead of the Trapezoidal Rule?
3. How would you modify the approach if the integral had different limits or a different function?
4. Can you explain why Simpson's Rule gives better accuracy compared to other numerical methods?
5. How does the symmetry of the function $f(x) = \ln(6 + 5\cos(x))$ influence the computation?

Tip: When using numerical integration methods like Simpson's Rule, always verify the result by increasing $n$ until the results stabilize within the desired accuracy.