Line integral $\int_K \frac{y}{x^2+y^2} dx -\frac{x}{x^2+y^2} dy$ Solve $\int_K \frac{y}{x^2+y^2} dx -\frac{x}{x^2+y^2} dy$ =? 
K: $x^2+y^2=1$ is oriented positively.

My attempt:
$\int_K \frac{y}{x^2+y^2} dx -\frac{x}{x^2+y^2} dy = |x=cost, y=sint,x'=- sint,y'=cost | $ = *
I put for $dx = -sint,dy=cost$
$= \int_0^{2\pi} \frac{sint}{cost^2+sint^2} -sint -\frac{cost}{cost^2+sint^2} cost  dt=  -\int_0^{2\pi} 1 dt = - 2 \pi $
We can't use Green's theorem because $P(x,y)$ and $Q(x,y)$ aren't continuous and their diveratives aren't continuous at point. $(0,0)$?

Another approach is
$$\frac{y}{x^2+y^2} dx -\frac{x}{x^2+y^2} dy=-\dfrac{x\,dy-y\,dx}{x^2+y^2}=-d\left(\arctan\dfrac{y}{x}\right)$$
so
$$\int_K \frac{y}{x^2+y^2} dx -\frac{x}{x^2+y^2} dy=-\int_K  d\left(\arctan\dfrac{y}{x}\right)=-\int_0^{2\pi}dt=-2\pi$$
