1. **Problem:** Evaluate the integral $$\int x \sin x \, dx$$ using integration by parts. 2. **Formula:** Integration by parts states: $$\int u \, dv = uv - \int v \, du$$ Choose: $$u = x \implies du = dx$$ $$dv = \sin x \, dx \implies v = -\cos x$$ 3. **Apply formula:** $$\int x \sin x \, dx = -x \cos x + \int \cos x \, dx$$ 4. **Integrate remaining integral:** $$\int \cos x \, dx = \sin x$$ 5. **Final answer:** $$\int x \sin x \, dx = -x \cos x + \sin x + C$$