Comparing it with $Mdx+Ndy=0$

We get

$$M=y\Bigg(1+\dfrac 1x\Bigg) +\cos y \space \& \space N=x+\log x-x\sin y$$

$\therefore \dfrac {\partial M}{\partial y}=1+\dfrac 1x-\sin y\\ \dfrac {\partial N}{\partial x}=1+\dfrac 1x-\sin y\\ \dfrac {\partial M}{\partial y}=\dfrac {\partial N}{\partial x} \text{ which is exact }\\ \therefore \int Mdx=\int \Bigg(y+\dfrac yx+\cos y\Bigg)dx\\ =yx+y\log x+x\cos y\\ \int (\text{ N terms free from x}) dy=\int0dy\\ =0\\ \therefore \text{ The solution is }\\ yx+y\log x+x\cos y=c$