To apply Simpson's Rule for the given integral:

$I = \int_0^2 \frac{e^x}{1 + x^2} \, dx$

with $n = 10$ (where $n$ must be even), let's break it down step by step.

Simpson's Rule Formula

Simpson's Rule for approximating integrals is given by:

$I \approx \frac{h}{3} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-1}) + f(x_n) \right]$

Where:

• $h = \frac{b - a}{n}$ is the step size.
• $x_0 = a$, $x_n = b$, and intermediate points are $x_1, x_2, \ldots, x_{n-1}$.
• $f(x)$ is the function we are integrating.

Step-by-Step Process

1. Function: $f(x) = \frac{e^x}{1 + x^2}$

2. Interval: $a = 0, \quad b = 2$

3. Step Size: With $n = 10$, the step size $h$ is: $h = \frac{2 - 0}{10} = 0.2$

4. Points: The points $x_i$ are calculated as: $x_0 = 0, \quad x_1 = 0.2, \quad x_2 = 0.4, \quad \dots, \quad x_9 = 1.8, \quad x_{10} = 2$

5. Apply Simpson's Rule: Now, calculate the values of $f(x_i)$ at each of these points and apply the Simpson's Rule formula.

Let me calculate this for you.The approximate value of the integral using Simpson's Rule with $n = 10$ is:

$I \approx 2.663$

Would you like any more details or further clarifications?

Here are some related questions to consider:

1. How does Simpson's Rule compare with other numerical methods like the Trapezoidal Rule?
2. What is the error bound in Simpson's Rule?
3. How does the choice of $n$ affect the accuracy of Simpson's Rule?
4. Can Simpson's Rule be used for integrals with singularities?
5. How would the approximation change if $n$ were different?

Tip: Simpson's Rule is most accurate when the function being integrated is smooth and well-approximated by parabolic segments.