https://file.aiquickdraw.com/mathbot/file/1727176932383f87vstzi.jpg

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.

Use Simpson's rule to approximate the integral \( \int_0^2 \frac{e^x}{1+x^2} \, dx \) with n = 10.

Learn how to approximate the integral of f(x) = e^x / (1 + x^2) from 0 to 2 using Simpson's Rule with n = 10. This step-by-step solution covers numerical integration techniques and the application of Simpson's Rule.

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