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.