To solve the integral \(\int e^{-x} \, dx\), we follow these steps:

1. Recognize that \(e^{-x}\) is an exponential function, and its integral is a standard form.
2. Use the rule that the integral of \(e^{ax}\) with respect to \(x\) is \(\frac{1}{a}e^{ax} + C\), where \(a\) is a constant and \(C\) is the constant of integration.

In this case, \(a = -1\). Therefore, we apply the rule:

\[
\int e^{-x} \, dx = \frac{1}{-1}e^{-x} + C = -e^{-x} + C
\]

So, the integral \(\int e^{-x} \, dx\) evaluates to:

\[
-e^{-x} + C
\]

where \(C\) is the constant of integration.

Would you like further details or have any questions? Here are some related questions to explore:

1. How do you integrate \(e^{2x}\)?
2. What is the integral of \(e^{-2x}\)?
3. How do you find the integral of \(e^{x^2}\)?
4. Can you explain the rule for integrating \(e^{ax + b}\)?
5. How do you integrate \(\int x e^{-x} \, dx\)?
6. What is the substitution method for integrating exponential functions?
7. How do you integrate \(e^{-x^2}\)?
8. What are the applications of integrating exponential functions?

**Tip:** When integrating exponential functions, always check if substitution can simplify the integral.

Learn how to integrate e^(-x) with a detailed step-by-step solution. This problem covers the integration of exponential functions, specifically e^(-x), using the integral formula. Suitable for advanced high school and college-level students.

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