You provided an integral $\int_0^2 \frac{e^x}{1+x^2} dx$ with $n = 10$. You are likely asking to solve this integral numerically using three methods: rectangular (midpoint), trapezoidal, and Simpson's rule, with 10 subdivisions ($n = 10$).

I will first break down how each method works and then calculate the result for the given integral:

1. Rectangular Rule (Midpoint Method): Approximates the integral by summing the areas of rectangles, where the height is determined by the function value at the midpoint of each subinterval.
2. Trapezoidal Rule: Approximates the integral by summing the areas of trapezoids under the curve.
3. Simpson’s Rule: Combines parabolic arcs through pairs of points, offering more accuracy than the other two methods.

Let me now compute these three approximations for the given integral.The numerical approximations for the integral $\int_0^2 \frac{e^x}{1+x^2} dx$ using $n = 10$ subdivisions are as follows:

1. Rectangular Rule (Midpoint Method): $2.6644$
2. Trapezoidal Rule: $2.6608$
3. Simpson's Rule: $2.6632$

Would you like further clarification on these methods or more details on the calculations?

Here are some related questions for deeper exploration:

1. How does increasing the number of subdivisions ($n$) improve the accuracy of each method?
2. Why does Simpson's rule generally provide a more accurate result compared to the rectangular and trapezoidal rules?
3. What are the limitations of using these numerical methods for more complex integrals?
4. How would the results change if the integral limits were different?
5. Can these methods be applied to improper integrals, and if so, how?

Tip: Simpson's rule works best when the function being integrated is smooth and can be well-approximated by parabolas over small intervals.