Integrating factor depending on one coordinate (${\mu }^{\prime }/\mu =(P_y-Q_x)/Q$)

For a differential one-form $Pdx+Qdy$, seek a nonzero integrating factor $\mu (x)$. Expanding $(\mu P{)}_y=(\mu Q{)}_x$ gives the displayed equation where $Q\ne 0$. Such a factor exists locally when $(P_y-Q_x)/Q$ depends only on $x$, and is $\mu =\exp \int [(P_y-Q_x)/Q]dx$. Likewise, a factor depending only on $y$ obeys ${\mu }^{\prime }/\mu =(Q_x-P_y)/P$ where $P\ne 0$. The resulting product is an exact differential form on a sufficiently small regular domain; singular lines of the factor need separate treatment.