[Math] How to integrate this function with cos and sin

Question

I'm trying to do the following:

$$\int \cos(wt)w\sin(wt+\phi) \,\mathrm{d}t$$

I can't figure what to do here. I can't do integration by parts because of the $\phi$. Any ideas?

Answer 1

FWIW

$$\eqalign{ & \int {w\cos wt\sin \left( {wt + \phi } \right)dt} = \cr & \int {w\cos wt\left( {\sin wt\cos \phi + \cos wt\sin \phi } \right)dt} = \cr & \cos \phi \int {w\cos wt\sin wtdt} + \sin \phi \int {w{{\cos }^2}wtdt} = \cr & \frac{{\cos \phi }}{2}\int {w\sin 2wtdt} + \frac{{\sin \phi }}{2}\int {w\left( {1 + \cos 2wt} \right)dt} = \cr & w\left\{ {\frac{{\sin \phi }}{2}\left( {\frac{{2t + \sin 2wt}}{2}} \right) - \frac{{\cos \phi }}{2}\left( {\frac{{\sin 2wt}}{2}} \right)} \right\} + C \cr} $$