There are a number of typographical errors in your post (at a glance, there's two dx's and a missing term in the transformation Jacobian).
From what I can tell, the region of integration is the upper half of the circle
between the line x = R/2 and the y-axis. Do you realise this?
A transformation is then made to polar coordinates. Do you realise that? Are you familiar with them?
Do you know how to describe the region of integration using polar coordinates? In particular:
1. Do you see how to express the line x = R/2 and the circle in polar coordinates.
2. Do you see how to get the integral terminals for the angle?