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The integral given in the image is:

\[
I = \int_0^2 \frac{e^x}{1 + x^2} \, dx, \quad n = 10
\]

It seems like you are asking for a numerical approximation of this definite integral, possibly using numerical methods (such as trapezoidal rule or Simpson's rule) since \(n = 10\) might represent the number of subintervals.

Let's calculate the integral numerically with \( n = 10 \) subintervals. I will compute the result for you.The numerical approximation of the integral 

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

is approximately \( 2.663 \), with a very small error margin of \( 3.02 \times 10^{-10} \).

Would you like more details or clarification on any step?

Here are five related questions to consider:
1. How would this integral change if the limits were from 0 to infinity?
2. What method can be used if \(n = 10\) refers to using a midpoint or trapezoidal rule?
3. How does \(e^x\) affect the behavior of the integral as \(x\) increases?
4. What happens if we replace \(1 + x^2\) in the denominator with another polynomial?
5. How can we estimate the error when using different numerical integration methods?

**Tip:** In numerical integration, choosing an appropriate method (like Simpson’s rule or trapezoidal rule) depends on the function’s smoothness over the interval.

Evaluate the integral ∫(e^x) / (1 + x^2) dx from 0 to 2, with n = 10.

This problem involves numerically approximating the integral ∫(e^x)/(1 + x^2) dx from 0 to 2, using n=10 subintervals. Learn how to apply methods like the trapezoidal rule or Simpson's rule to solve this integral.

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