Let $\mu(x,y)$ be an integrating factor of the differential equation $P(x,y)\,dx+Q(x,y)\,dy=0$. I already showed that we have $$\frac{\partial P}{\partial y}-\frac{\partial Q}{\partial x}=Q\frac{\partial}{\partial x} \log|\mu| - P\frac{\partial}{\partial y} \log|\mu|.$$ Now I have to deduce that if $(\partial P/\partial y-\partial Q/\partial x)/Q$ is a function of $x$ alone, say $f(x)$, then the function $e^{\int f(x) \, dx}$ is an integrating factor of the equation.
I couldn't find the way to see this.
The previous exercise shows that the equation $y'+P(x)y=Q(x)$ has the integrating factor $e^{\int P(x)\,dx}$, but I don't know if I can use it here.