1. The problem is to find the indefinite integral of the function $e^x$, which is written as $\int e^x \, dx$. 2. The formula for the integral of the exponential function $e^x$ is: $$\int e^x \, dx = e^x + C$$ where $C$ is the constant of integration. 3. This formula works because the derivative of $e^x$ is $e^x$, so integrating $e^x$ returns the original function plus a constant. 4. Applying the formula directly: $$\int e^x \, dx = e^x + C$$ 5. Therefore, the solution to the integral is $e^x + C$. This means the antiderivative of $e^x$ is $e^x$ itself, plus an arbitrary constant to account for all possible antiderivatives.