Integrate$ \int x \cos(x) \sin(x)\; dx$ 
Integrate: $$\int x \cos(x) \sin(x) \;dx$$

I've been trying to integrate by parts, but I can't! I know there's a trigonometric function about $\cos (2x)$ but I don't know how to integrate that function with $\sin(x)$. 

Given $$\int x\cos (x)\sin(x)\ dx$$
Use the identity $\cos(x)\sin(x)=\dfrac{\sin(2x)}{2}$
Now $$\int x\ \dfrac{\sin(2x)}{2}\ dx=\dfrac12\int x\sin(2x)\ dx$$
Now use Integration By Parts $u=x$ and $v^{\prime}=\sin(2x)$
Can you continue from here?
