= Solution
For an <exact differential> $Pdx+Qdy$, equality of mixed <partial derivatives> requires $P_y=Q_x$. Here this gives
$$
2\mu\cos y=-(x\mu'+\mu)\cos y.
$$
On an open two-dimensional domain the coefficient relation is $x\mu'+3\mu=0$, so $\mu=Cx^{-3}$ on a connected interval avoiding zero. The normalization gives $C=1$ on the positive interval through $x=1$. Integrating $f_x=2x^{-3}\sin y$ and checking $f_y$ yields
$$
\boxed{\mu(x)=x^{-3},\qquad f(x,y)=-\frac{\sin y}{x^2}+C_0,\quad x>0.}
$$
There is no smooth extension of this normalized integrating factor across $x=0$; an independently specified factor on $x<0$ is not fixed by data at one.
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