1. The problem is to find the indefinite integral $\int \sin 2x \, dx$. 2. Recall the integral formula: $\int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C$ where $a$ is a constant. 3. Here, $a = 2$, so substitute into the formula: $$\int \sin 2x \, dx = -\frac{1}{2} \cos 2x + C$$ 4. Explanation: We use a substitution mindset here without formally substituting. Since derivative of $\cos 2x$ is $-2 \sin 2x$, the integral of $\sin 2x$ must include a factor $-\frac{1}{2}$ to cancel out the 2 when differentiating. 5. Therefore, the final answer is: $$\boxed{-\frac{1}{2} \cos 2x + C}$$