let $F=\left(\frac{-y}{x^2+y^2},\frac{x}{x^2+y^2}\right)$ and $R(t)=(\cos t,\sin t)$ (the curve is a circle with radius 1)
now: \begin{equation} \int_{R}F_1.dx+F_2.dy = \int_{0}^{2\pi}-\sin t\ dt + \int_{0}^{2\pi}\cos t\ dt = 0 \end{equation} but the book concluded from the Green's theorem that the answer is $2\pi$ (the book take a neighborhood near $0$ (because $0$ is not in the domain) and ...)
so why my answer is wrong and can you give me a solution because I can't understand the solution in the book
The vector field on the circle is $$ F=(F_1,F_2)=(-\sin t,\cos t). $$ Since $(x,y)=(\cos t,\sin t)$ you should also take into account that $$ (dx,dy)=(x'dt,y'dt)=(-\sin t\,dt,\cos t\,dt). $$ With $$ F_1\,dx+F_2\,dy=(\sin^2t+\cos^2t)\,dt=dt $$ it makes a simple integration.