We are tasked with evaluating the following definite integral:

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

This integral doesn't have a straightforward antiderivative, but it can be approached numerically.

I will compute the value of this integral using numerical methods. Let's proceed with that.The value of the integral

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

is approximately \( 1.3133 \), with an extremely small numerical error of \( 1.46 \times 10^{-14} \).

Would you like to explore the details of this calculation or have any questions?

Here are 5 related questions you might find interesting:
1. How can we approximate integrals using numerical methods like Simpson's rule or trapezoidal rule?
2. What conditions determine if an integral has a closed-form solution or not?
3. How do exponential functions influence the behavior of an integral?
4. What are some techniques for evaluating integrals involving exponential functions analytically?
5. How does the integral behave as we extend the limits beyond [0,1]?

**Tip:** For integrals with no simple antiderivative, numerical integration is often the most practical approach.

Explore the solution of the definite integral \( I = \int_0^1 \frac{1 + e^x}{x + e^x} \, dx \). This problem involves exponential functions and numerical integration methods. The approximate result of the integral is 1.3133 with a very small error. Learn about numerical integration techniques and the conditions for closed-form solutions.

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