To evaluate the integral:

$I = \int \frac{4}{e^{x} \sqrt{2 + e^{2x}}} \, dx$

We can proceed with a substitution to simplify the integral.

Step 1: Let $u = e^x$

Then, $du = e^x \, dx$, and $e^x = u$.

This transforms the integral as follows:

$I = \int \frac{4}{u \sqrt{2 + u^2}} \, du$

Step 2: Simplify the integral

We now have:

$I = 4 \int \frac{1}{u \sqrt{2 + u^2}} \, du$

This is a standard integral, which can be solved by the substitution $v = 2 + u^2$, leading to:

$dv = 2u \, du$

Step 3: Apply the substitution

Substitute into the integral:

$I = 4 \int \frac{1}{2 \sqrt{v}} \, dv$

This simplifies to:

$I = 2 \int v^{-1/2} \, dv$

Step 4: Integrate

The integral of $v^{-1/2}$ is $2v^{1/2}$, so we have:

$I = 2 \cdot 2v^{1/2} = 4v^{1/2}$

Step 5: Substitute back $v = 2 + u^2$

Now, substitute back for $v$:

$I = 4 \sqrt{2 + u^2}$

Step 6: Substitute back $u = e^x$

Finally, substitute $u = e^x$ back into the expression:

$I = 4 \sqrt{2 + e^{2x}} + C$

Thus, the integral is:

$\int \frac{4}{e^{x} \sqrt{2 + e^{2x}}} \, dx = 4 \sqrt{2 + e^{2x}} + C$

Let me know if you'd like more details on any step or have other questions! Here are a few related questions you might find useful:

1. What are some techniques for integrating rational functions involving exponentials and square roots?
2. How does the substitution $u = e^x$ work in simplifying integrals?
3. Can you solve integrals involving higher powers of exponentials, like $e^{3x}$ or $e^{4x}$?
4. What are the key steps in solving integrals using trigonometric substitutions?
5. How do you evaluate indefinite integrals that involve square roots of quadratic expressions?

Tip: When dealing with integrals involving exponentials and square roots, substitutions often work well, especially when you can recognize a pattern like $e^{2x}$ or other common forms.