The integral you provided is:

$\int \frac{e^{-x}}{(1 + e^{-x})^3} \, dx$

To solve this integral, we can proceed with a substitution that simplifies the expression. Let's set:

$u = 1 + e^{-x}$

Then, differentiate $u$ with respect to $x$:

$\frac{du}{dx} = -e^{-x} \Rightarrow du = -e^{-x} \, dx \Rightarrow -du = e^{-x} \, dx$

Substituting into the integral:

$\int \frac{e^{-x}}{(1 + e^{-x})^3} \, dx = \int \frac{1}{u^3} (-du) = -\int u^{-3} \, du$

Now we can integrate:

$-\int u^{-3} \, du = \frac{u^{-2}}{2} + C = \frac{1}{2u^2} + C$

Substituting back $u = 1 + e^{-x}$:

$\frac{1}{2(1 + e^{-x})^2} + C$

Therefore, the final result is:

$\boxed{\frac{1}{2(1 + e^{-x})^2} + C}$

where $C$ is the constant of integration.

Would you like further details or have any questions?

Here are 5 related questions for further exploration:

1. How would the integral change if the power of the denominator were different, such as $(1 + e^{-x})^n$?
2. How do substitutions help in solving integrals that involve exponential functions?
3. What would the integral be if there were an $e^x$ in the numerator instead of $e^{-x}$?
4. How can this method of substitution be applied to solve other integrals involving $e^{-x}$?
5. What is the significance of the constant of integration in indefinite integrals?

Tip: When dealing with integrals involving exponentials, consider substitutions that simplify the exponential expression. It often transforms the integral into a more manageable form.