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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?

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### 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.

Explore the step-by-step solution of the integral ∫√(1 + e^x) dx using substitution methods and numerical techniques. This problem involves exponential functions and square roots, often requiring advanced techniques or numerical approaches for a complete solution.

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