[SOLVED] Check out my result : line integral

$\displaystyle I=\int_C \frac{dx+dy}{x^2+y^2}$ where $\displaystyle C$ is described by $\displaystyle r(t)=(\cos t, \sin t)$, $\displaystyle 0\leq x \leq 2\pi$.

My attempt : $\displaystyle |r'(t)|=1$ so $\displaystyle I=\int_0^{2\pi} \frac{\cos t dt+\sin tdt}{\cos^2t + \sin^2t}=0$.

I realize that $\displaystyle C$ is a circle of radius $\displaystyle 1$ and that if my result is good then $\displaystyle F(x,y)=\frac{dx+dy}{x^2+y^2}$ is a conservative field. But I'm not sure I've done it right.