1. The problem is to find the indefinite integral of the function $6e^x$ with respect to $x$. 2. Recall the integral formula for the exponential function: $$\int e^x \, dx = e^x + C$$ where $C$ is the constant of integration. 3. Since the integrand is $6e^x$, a constant multiple of $e^x$, we use the rule: $$\int a f(x) \, dx = a \int f(x) \, dx$$ where $a$ is a constant. 4. Applying this rule, we have: $$\int 6e^x \, dx = 6 \int e^x \, dx$$ 5. Substitute the integral of $e^x$: $$= 6 (e^x + C) = 6e^x + 6C$$ 6. Since $6C$ is just a constant, we write the final answer as: $$\boxed{6e^x + C}$$ This is the antiderivative of $6e^x$.