Solution of the differential equation $(\sin y+y(1+\frac{1}{x}\left)\right)dx+\left(x\right(1+\cos y)+\mathcal{l}nx)dy=0$ is

$(\sin y+y(1+\frac{1}{x}))dx+\left(x\right(1+\cos y)+\mathcal{l}nx)dy=0$

$\Rightarrow \sin ydx+x\cos ydy+xdy+ydx+\frac{y}{x}dx+\mathcal{l}nxdy=0$

$\Rightarrow d(x.\sin y)+d\left(xy\right)+d(y.\mathcal{l}nx)=0$

$\Rightarrow x.\sin y+xy+y.\mathcal{l}nx=c$