1. **State the problem:** We need to find the integral of the function $x \cos(2x)$ with respect to $x$. 2. **Formula and method:** We will use integration by parts, which states: $$\int u \, dv = uv - \int v \, du$$ 3. **Choose parts:** Let $$u = x \implies du = dx$$ $$dv = \cos(2x) dx \implies v = \frac{\sin(2x)}{2}$$ 4. **Apply integration by parts:** $$\int x \cos(2x) dx = x \cdot \frac{\sin(2x)}{2} - \int \frac{\sin(2x)}{2} dx$$ 5. **Integrate remaining integral:** $$\int \sin(2x) dx = -\frac{\cos(2x)}{2}$$ So, $$\int \frac{\sin(2x)}{2} dx = \frac{1}{2} \cdot \left(-\frac{\cos(2x)}{2}\right) = -\frac{\cos(2x)}{4}$$ 6. **Combine results:** $$\int x \cos(2x) dx = \frac{x \sin(2x)}{2} + \frac{\cos(2x)}{4} + C$$ 7. **Final answer:** $$\boxed{\int x \cos(2x) dx = \frac{x \sin(2x)}{2} + \frac{\cos(2x)}{4} + C}$$