1. **State the problem:** We need to find 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 $u = x$ and $dv = \sin x \, dx$. 3. **Compute derivatives and integrals:** $$du = dx$$ $$v = -\cos x$$ 4. **Apply integration by parts:** $$\int x \sin x \, dx = -x \cos x - \int -\cos x \, dx = -x \cos x + \int \cos x \, dx$$ 5. **Integrate remaining integral:** $$\int \cos x \, dx = \sin x$$ 6. **Combine results:** $$\int x \sin x \, dx = -x \cos x + \sin x + C$$ 7. **Final answer:** $$\boxed{-x \cos x + \sin x + C}$$