From what I understand about your problem, you're integrating $$\int\;\vec{F}\cdot d\vec{s} = \iint\;\vec{F}\cdot \vec{n} \; dx dy$$ on a circle of radius 1.
So basically the only thing you need to do is to express your vector $d\vec{s}$ in the easiest way. For example, in this case, you're dealing with a circle which is best dealt with in polar coordinates. Hence, you have for example $$\vec{n}\; dxdy = \pm r\cdot(-\sin \theta, \cos \theta) \;r \,drd\theta,$$ depending on in which direction the contour on your circle is oriented.
In polar coordinates, $\vec{F}$ reads $r(\sin \theta,\cos \theta)$. You put evrything in your integral and you integrate on $\theta$ and $r$ (the latter is constant) on the circle. It shouldn't cause you problems now.