1. **State the problem:** Calculate the integral $$\int x \sin x \, dx$$. 2. **Formula and method:** Use integration by parts, which states: $$\int u \, dv = uv - \int v \, du$$ Choose: $$u = x \quad \Rightarrow \quad du = dx$$ $$dv = \sin x \, dx \quad \Rightarrow \quad v = -\cos x$$ 3. **Apply integration by parts:** $$\int x \sin x \, dx = -x \cos x - \int -\cos x \, dx = -x \cos x + \int \cos x \, dx$$ 4. **Integrate remaining integral:** $$\int \cos x \, dx = \sin x$$ 5. **Combine results:** $$\int x \sin x \, dx = -x \cos x + \sin x + C$$ 6. **Final answer:** $$\boxed{-x \cos x + \sin x + C}$$