The given integral is:

$\int \sqrt{1 + e^x} \, dx$

This is not a standard integral, so we will need to use a substitution method or explore an approximation. Let’s proceed with substitution to simplify this expression.

Step-by-Step Solution:

1. Substitution: Let: $u = 1 + e^x$ Then: $du = e^x \, dx$ From the substitution $u = 1 + e^x$, we have $e^x = u - 1$, and so $du = (u - 1) \, dx$.

2. Rewrite the integral: The original integral becomes: $\int \sqrt{u} \cdot \frac{du}{u - 1}$

3. At this point, further simplification or numerical methods could be applied depending on the problem context, as solving this directly requires complex integration techniques, or it could lead to a numerical approximation.

The result will involve a more advanced method of integration (non-elementary). It can be approached through numerical approximation, or solved using specialized functions like hyperbolic functions.

Would you like to explore a numerical approach for an approximation, or look into advanced integration techniques?

Follow-up Questions:

1. How can you approach integrals involving square roots and exponentials?
2. What are the general steps for substitution in integration?
3. Can numerical methods be used to approximate non-elementary integrals?
4. How do special functions (like hyperbolic or elliptic functions) help solve complex integrals?
5. Why are some integrals non-elementary, and what are the implications for solving them?

Tip:

For integrals involving both exponentials and square roots, substitution is often the first step, but recognizing non-elementary integrals helps in choosing whether to use numerical methods or advanced functions.