The solution of $\left(y\left(1+x^{-1}\right)+\sin y\right)dx+\left(x+{\log }_{e}x+x\cos y\right)dy=0$  is

$\begin{array}{l}ydx+\frac{y}{x}\cdot dx+\sin ydx+xdy+{\log }_{e}xdy+x\cos ydy=0\\ \Rightarrow (ydx+xdy)+\left(y\cdot \frac{dx}{x}+{\log }_{e}xdy\right)+(\sin ydx+x\cos ydy)=0\\ \Rightarrow d\left(xy\right)+d\left(y{\log }_{e}x\right)+d(x\sin y)=0 integrating on both sides\\ xy+y{\log }_{e}x+x\sin y=C\end{array}$