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