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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.

Evaluate the integral \( \int \frac{e^{-x}}{(1 + e^{-x})^3} \, dx \).

Learn how to evaluate the integral \( \int \frac{e^{-x}}{(1 + e^{-x})^3} \, dx \) using substitution. This calculus problem involves exponential functions and the substitution method, suitable for university-level students.

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