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The exercise asks to approximate the integral 

\[
\int_2^5 x e^{-x^2} \, dx
\]

using both the **trapezoidal rule** and **Simpson's rule** for different values of \( n \) (number of subintervals). Here's how we can proceed:

### 1. **Trapezoidal Rule**:
The formula for the trapezoidal rule with \( n \) subintervals is:

\[
T_n = \frac{b - a}{2n} \left[ f(a) + 2 \sum_{i=1}^{n-1} f(x_i) + f(b) \right]
\]

where \( a = 2 \), \( b = 5 \), and \( f(x) = x e^{-x^2} \). 

### 2. **Simpson’s Rule**:
For Simpson’s rule, the formula is:

\[
S_n = \frac{b - a}{3n} \left[ f(a) + 4 \sum_{i \text{ odd}} f(x_i) + 2 \sum_{i \text{ even}} f(x_i) + f(b) \right]
\]

This method only works if \( n \) is even.

### Steps:
- For each value of \( n = 4, 6, 10, 12 \), we will calculate the approximations using both methods.
- First, I'll calculate the approximations for \( n = 4 \), and we can continue for the other values.

Let me calculate this now.Here are the results of approximating the integral \(\int_2^5 x e^{-x^2} \, dx\) using the trapezoidal rule and Simpson's rule for different values of \(n\):

| Number of Subintervals (n) | Trapezoidal Rule (RT) | Simpson's Rule (RS) |
|----------------------------|-----------------------|---------------------|
| 4                          | 0.01482               | 0.01060             |
| 6                          | 0.01176               | 0.00946             |
| 10                         | 0.01011               | 0.00919             |
| 12                         | 0.00982               | 0.00917             |

Would you like further details or have any questions?

### Related Questions:
1. How does the accuracy improve as the number of subintervals increases?
2. Why is Simpson's rule generally more accurate than the trapezoidal rule for smooth functions?
3. How can you derive the formula for the trapezoidal rule?
4. Can you explain the geometric interpretation of Simpson’s rule?
5. How do we evaluate the error in numerical integration methods like these?

#### Tip:
Simpson's rule tends to provide better accuracy because it uses quadratic interpolation, which fits the curve of the function more closely than the linear interpolation used by the trapezoidal rule.

Use the trapezoidal rule and Simpson's rule to approximate the integral \( \int_2^5 x e^{-x^2} \, dx \) with the values of n = 4, 6, 10, 12.

This page explains how to approximate the integral \( \int_2^5 x e^{-x^2} \, dx \) using the trapezoidal rule and Simpson's rule for n = 4, 6, 10, and 12. Learn the key concepts behind numerical integration and how different methods improve accuracy.

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