1. **State the problem:** We need to find the integral of $\cos 2x$ with respect to $x$. 2. **Recall the formula:** The integral of $\cos(ax)$ is $\frac{1}{a} \sin(ax) + C$, where $a$ is a constant and $C$ is the constant of integration. 3. **Apply the formula:** Here, $a = 2$, so $$\int \cos 2x \, dx = \frac{1}{2} \sin 2x + C$$ 4. **Explanation:** We divide by the coefficient of $x$ inside the cosine function to account for the chain rule in reverse. 5. **Final answer:** $$\int \cos 2x \, dx = \frac{1}{2} \sin 2x + C$$