1. The problem is to evaluate the integral of the function $e^x$ from $x=0$ to $x=-\infty$. 2. The integral of $e^x$ with respect to $x$ is given by the formula: $$\int e^x \, dx = e^x + C$$ where $C$ is the constant of integration. 3. Since this is a definite integral, we evaluate: $$\int_0^{-\infty} e^x \, dx = \lim_{a \to -\infty} \int_0^a e^x \, dx$$ 4. Calculate the integral: $$\int_0^a e^x \, dx = e^x \Big|_0^a = e^a - e^0 = e^a - 1$$ 5. Take the limit as $a$ approaches $-\infty$: $$\lim_{a \to -\infty} (e^a - 1) = 0 - 1 = -1$$ 6. Therefore, the value of the integral is: $$\int_0^{-\infty} e^x \, dx = -1$$ Note: The integral is taken from a higher limit to a lower limit, so the negative sign appears. If we reverse the limits, the integral would be positive 1.