1. **State the problem:** We need to find the integral of the function $e^{-x}$ with respect to $x$. 2. **Recall the formula:** The integral of $e^{ax}$ with respect to $x$ is given by $$\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C,$$ where $a$ is a constant and $C$ is the constant of integration. 3. **Apply the formula:** Here, $a = -1$, so $$\int e^{-x} \, dx = \frac{1}{-1} e^{-x} + C = -e^{-x} + C.$$ 4. **Explain:** The negative sign appears because the exponent has a coefficient of $-1$. When integrating, we divide by this coefficient. 5. **Final answer:** $$\boxed{-e^{-x} + C}.$$