Check whether integral of form: $\frac{y}{x^2+xy+y^2}dx+\frac{-x}{x^2+xy+y^2}dy$ on $R^2$ \ {($0,0$)} is path independent.
The integral of form will be path independent if there exist twice continuously differentiable function $h$ such that partial derivative of $h$ with respect to $x$ is equal to $\frac{y}{x^2+xy+y^2}$ and partial derivative of $h$ with respect to $y$ is equal to $\frac{-x}{x^2+xy+y^2}$ right?
I think such function is $$h(x,y)=\frac{2\sqrt3}{3}\arctan(\frac{\frac{2\sqrt3}{3}x+\frac{\sqrt3}{3}y}{y})$$
So integral of this form is path independent.
However if we consider path $u(t)=(\cos t \sin t)$ ,$t \in [-\pi,\pi]$ then integral of our form is equal to: $$-\int_{-\pi}^{\pi}\frac{1}{1+\sin t\cos t}dt$$ which is non zero but the path is closed so we don't have path independence.
What is wrong?