1. **Stating the problem:** We want to find the integral $$\int x \cos x \, dx$$. 2. **Formula and method:** We use integration by parts, which states: $$\int u \, dv = uv - \int v \, du$$ Choose: $$u = x \implies du = dx$$ $$dv = \cos x \, dx \implies v = \sin x$$ 3. **Apply integration by parts:** $$\int x \cos x \, dx = x \sin x - \int \sin x \, dx$$ 4. **Integrate remaining integral:** $$\int \sin x \, dx = -\cos x$$ 5. **Combine results:** $$\int x \cos x \, dx = x \sin x - (-\cos x) + C = x \sin x + \cos x + C$$ 6. **Final answer:** $$\boxed{\int x \cos x \, dx = x \sin x + \cos x + C}$$