1. **State the problem:** We need to find the integral of the function $2 \sin(2x)$. 2. **Recall the formula:** The integral of $\sin(ax)$ with respect to $x$ is $-\frac{1}{a} \cos(ax) + C$, where $a$ is a constant and $C$ is the constant of integration. 3. **Apply the formula:** Here, $a = 2$, so $$\int 2 \sin(2x) \, dx = 2 \int \sin(2x) \, dx = 2 \left(-\frac{1}{2} \cos(2x)\right) + C$$ 4. **Simplify:** $$= -\cos(2x) + C$$ 5. **Interpretation:** The integral of $2 \sin(2x)$ is $-\cos(2x) + C$, where $C$ is an arbitrary constant representing the family of antiderivatives. **Final answer:** $$\int 2 \sin(2x) \, dx = -\cos(2x) + C$$