1. The problem is to find the integral of $\cos 3x$ with respect to $x$, i.e., compute $\int \cos 3x \, dx$. 2. Recall the formula for integrating cosine of a linear function: $\int \cos(ax) \, dx = \frac{1}{a} \sin(ax) + C$, where $a$ is a constant and $C$ is the constant of integration. 3. Here, $a = 3$, so applying the formula: $$\int \cos 3x \, dx = \frac{1}{3} \sin 3x + C$$ 4. This means the antiderivative of $\cos 3x$ is $\frac{1}{3} \sin 3x$ plus the constant of integration. 5. Therefore, the final answer is: $$\boxed{\frac{1}{3} \sin 3x + C}$$