The line integral of ∮{y/(x²+y²)dx-x/(x²+y²)dy} around the unit circle around the origin traversed once in the counter clockwise =? 0 or -2π

if I put x=cosθ and y=sinθ ; dx=-sinθdθ, dy=cosθdθ where 0≤θ≤2π and evaluated the integral I will get -2π as the answer.

However if I evaluated the integral using Green's theorem I will get 0 as the answer.

I also checked the curl of the vector f= y/(x²+y²)i-x/(x²+y²)j which is equal to zero. That implies the line integral around the unit circle should be zero as the vector filed is irrotational.

My question is why I'm getting -2π as the answer when i put x=cosθ and y=sinθ ; dx=-sinθdθ, dy=cosθdθ where 0≤θ≤2π and evaluated the integral?

Can anybody answer the question? Thanks in advance.