To evaluate the integral $I = \int_0^1 (x^e + e^x) \, dx$, we will compute each part separately:

1. Integral of $x^e$

The first part of the integral is $\int_0^1 x^e \, dx$.

To integrate $x^e$, we use the power rule: $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ where $n = e$, so: $\int x^e \, dx = \frac{x^{e+1}}{e+1} + C$

Evaluating this from 0 to 1: $\left[ \frac{x^{e+1}}{e+1} \right]_0^1 = \frac{1^{e+1}}{e+1} - \frac{0^{e+1}}{e+1} = \frac{1}{e+1}$

2. Integral of $e^x$

The second part of the integral is $\int_0^1 e^x \, dx$.

The integral of $e^x$ is straightforward: $\int e^x \, dx = e^x + C$

Evaluating this from 0 to 1: $\left[ e^x \right]_0^1 = e^1 - e^0 = e - 1$

3. Adding both parts

Now, summing both parts together: $I = \frac{1}{e+1} + (e - 1)$

Thus, the value of the integral is: $I = \frac{1}{e+1} + e - 1$

Would you like any further details or have any questions about this solution?

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Here are 5 related questions:

1. How do you integrate functions with exponents involving constants?
2. What happens when you change the limits of integration?
3. How can you evaluate integrals involving logarithmic functions?
4. What are some applications of definite integrals in real life?
5. Can you evaluate integrals with more complex transcendental functions?

Tip: When evaluating integrals, always split the integral into simpler parts if possible to make the process easier.