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The integral you provided is:

\[
\int \frac{dx}{\sqrt{1 + e^x}}
\]

### To solve this:
We can approach this by substitution. Let’s make the substitution:

\[
u = 1 + e^x
\]

Then, the derivative of \( u \) with respect to \( x \) is:

\[
du = e^x dx
\]

This implies that:

\[
dx = \frac{du}{e^x}
\]

Now, since \( u = 1 + e^x \), it follows that:

\[
e^x = u - 1
\]

Thus, \( dx = \frac{du}{u - 1} \), and the integral becomes:

\[
\int \frac{1}{\sqrt{u}} \cdot \frac{du}{u - 1}
\]

This expression is more complicated but can be handled through partial fractions or further transformations. Let me compute the next steps for you.The solution to the integral is:

\[
\log\left(\frac{1 - \frac{1}{\sqrt{1 + e^x}}}{1 + \frac{1}{\sqrt{1 + e^x}}}\right)
\]

This expression simplifies the result of the integration process. If you'd like further simplification or explanation of specific steps, feel free to ask.

Do you have any questions or would you like more details? Here are five related questions to deepen your understanding:

1. How do we perform substitution in more complex integrals?
2. What is the role of logarithmic functions in integration?
3. How can partial fractions help in solving integrals?
4. What are other common substitution techniques for integrals involving exponential functions?
5. How do we recognize integrals that can be solved by logarithmic methods?

**Tip:** When working with integrals involving exponential functions, try to identify expressions that can be transformed using substitutions to simpler forms, especially when logarithms or roots are involved.

Explore how to solve the integral ∫ dx / √(1 + e^x) using substitution and logarithmic techniques. This problem introduces key concepts in exponential and logarithmic integration, suitable for advanced high school and college-level students.

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