To find all antiderivatives of the given function $f(x) = e^{-18x}$, we compute the indefinite integral:

$\int f(x) \, dx = \int e^{-18x} \, dx$

Step 1: Recall the rule for exponential integrals

The integral of $e^{ax}$ is: \int e^{ax} \, dx = \frac{1}{a} e^{ax} + C \quad \text{(where a \neq 0)}.

Step 2: Apply the formula

Here, $a = -18$. Therefore: $\int e^{-18x} \, dx = \frac{1}{-18} e^{-18x} + C$

Final Answer

$\int f(x) \, dx = -\frac{1}{18} e^{-18x} + C$

where $C$ is the constant of integration.

Would you like a step-by-step breakdown of how this rule is derived or applied in specific scenarios?

---

5 Related Questions

1. What is the significance of the constant of integration in indefinite integrals?
2. How do you compute the definite integral of $f(x) = e^{-18x}$ over a specific interval?
3. Can this result be applied to solve real-world exponential decay problems?
4. What happens if the exponent has a more complex function, like $e^{-18x^2}$?
5. How does substitution help in evaluating integrals with composite exponential functions?

Tip

Always double-check the sign and coefficient when integrating exponential functions—they are common sources of errors!