To integrate the expression $a e^x + b e^x$, we can factor out $e^x$, as both terms contain this factor. The integration process becomes:

$\int (a e^x + b e^x) \, dx$

Factor out $e^x$:

$= \int e^x (a + b) \, dx$

Since $a + b$ is just a constant, you can take it outside of the integral:

$= (a + b) \int e^x \, dx$

The integral of $e^x$ is simply $e^x$, so we get:

$= (a + b) e^x + C$

where $C$ is the constant of integration.

Thus, the result is:

$\boxed{(a + b) e^x + C}$

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

Here are a few questions that might help expand on this topic:

1. What would happen if the integral had a different base, like $e^{2x}$?
2. How would you handle integrals involving constants that are not factored out, like $a e^x + b$?
3. Could you explain how to solve an integral like $\int e^{ax} \, dx$?
4. How does the method of integration change when there are products involving polynomials and exponentials?
5. What is the general rule for integrating exponential functions?

Tip: When you encounter sums in an integral, you can always split them up into separate integrals, as long as they are within the same limits of integration.