To integrate $\frac{e^x}{1 + e^{2x}}$, let's proceed with the following steps:

Step 1: Simplify the denominator

Notice that the denominator can be rewritten as: $1 + e^{2x} = 1 + (e^x)^2$

Thus, the integral becomes: $\int \frac{e^x}{1 + (e^x)^2} \, dx$

Step 2: Substitution

Let $u = e^x$, which implies $du = e^x \, dx$. Substituting this into the integral, we get: $\int \frac{du}{1 + u^2}$

Step 3: Solve the integral

The integral $\int \frac{1}{1 + u^2} \, du$ is a standard result: $\int \frac{1}{1 + u^2} \, du = \tan^{-1}(u)$

Step 4: Substitute back $u = e^x$

Now, replace $u$ with $e^x$ to obtain the final result: $\int \frac{e^x}{1 + e^{2x}} \, dx = \tan^{-1}(e^x) + C$

Thus, the solution to the integral is: $\boxed{\tan^{-1}(e^x) + C}$

Would you like a detailed explanation of any step?

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Here are 5 related questions for practice:

1. What is the integral of $\frac{1}{1 + x^2}$?
2. Integrate $\frac{x}{1 + x^2}$.
3. Find the derivative of $\tan^{-1}(e^x)$.
4. Evaluate $\int \frac{e^x}{1 + e^x} \, dx$.
5. Solve $\int \frac{e^{2x}}{1 + e^{4x}} \, dx$.

Tip: Substitution is a powerful tool when you can recognize expressions that simplify the integral into a standard form.