Vector Calculus · Hard problem

Let $\mathbf F = \left(\dfrac{-y}{x^2 + y^2},\; \dfrac{x}{x^2 + y^2}\right)$. Evaluate $\displaystyle\int_C\mathbf F\cdot d\mathbf r$, where $C$ is the straight line segment from $(1, 0)$ to $(-1, \sqrt 3)$. Give your answer in terms of $\pi$.