The integral we are tasked with solving is:

$\int \frac{e^x}{1 + e^{2x}} \, dx$

Step-by-Step Solution:

1. Substitution:

Let's try a substitution to simplify the expression. Let: $u = e^x$ Then, the derivative of $u$ with respect to $x$ is: $du = e^x \, dx$ This transforms the integral into: $\int \frac{du}{1 + u^2}$

2. Recognizing a Standard Integral:

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

3. Substitute Back:

Now, we substitute back $u = e^x$: $\int \frac{e^x}{1 + e^{2x}} \, dx = \tan^{-1}(e^x) + C$

Final Answer:

$\int \frac{e^x}{1 + e^{2x}} \, dx = \tan^{-1}(e^x) + C$ where $C$ is the constant of integration.

Do you want any further explanation or details about this solution?

Here are five follow-up questions to deepen understanding:

1. What is the purpose of using substitution in integrals?
2. How do you identify when to use the substitution method?
3. Why is the integral $\frac{1}{1 + u^2}$ linked to $\tan^{-1}(u)$?
4. What are some other integrals that result in inverse trigonometric functions?
5. How would the result change if the denominator were $1 + e^x$ instead of $1 + e^{2x}$?

Tip: Always check if the integrand can be simplified using a substitution, as it often turns a complex expression into a recognizable form.