many thanks! I will read it carefully, but it looks like it contains the answer

well, in my case it is a non-ramified covering of smooth manifolds

If I understand correctly, you should take the Riemann surface of this function in the complex 2-space with coordinates (x, y), and integrate the form dx/y along the path which lifts this interval to one branch until the branching point, then goes around this point and goes back along another lift of this interval, but on a different branch.

There is a detailed argument in this paper arxiv.org/pdf/1504.04405.pdf The Origins of Complex Geometry in the 19th Century (Raymond O. Wells, Jr.), Section 6