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.

Explore the integration of e^x / (1 + e^2x) step by step using substitution. This calculus problem demonstrates key integration techniques and results, such as the standard integral for 1 / (1 + u^2). Ideal for undergraduate calculus students.

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