Consider the differential form $$\omega(x, y) =\frac{8xy}{(4x^2+3)\left(ln^2\left( x^2+\frac34\right)+y^2\right)}dx -\frac{ln\left( x^2+\frac34\right)}{\left(ln^2\left( x^2+\frac34\right)+y^2\right)} dy.$$
Its domain is $\mathbb{R}^2\setminus\left\{(\pm\frac12, 0)\right\}$ and it is exact in its domain. Since $\mathbb{R}^2\setminus\left\{(\pm\frac12, 0)\right\}$ is not simply connected, I need to evaluate the circulation around the points $(\pm\frac12, 0)$. If it is $0$, we have an exact form.
My question is: it is needed to consider a circuit which wraps the points separately or I can take into account a circuit which wraps both?
Thank you in advance!
${\bf EDIT:}$ Could someone help me to verify that the circulation around each point is $0$? I just need a hint about the right closed curve to use (circle, rectangle, square, etc)
No, the circulation must vanish for all closed loops (smooth paths with the same start and end point). Just because it holds for some, in particular those which enclose both the singular points, is not sufficient to conclude it is exact.