I'm having some trouble with the following exercise:
Let $\alpha:[a,b]\to \mathbb R^2\setminus\{(0,0)\}$ be a curve of class $C^1$ and $\varphi:[a,b]\to \mathbb R$ be a $C^1$ function such that: $$\forall t\in [a,b], \alpha(t) = ||\alpha(t)||\cdot(\cos(\varphi(t)), \sin(\varphi(t)))$$ And let $C$ be the path represented by the curve $\alpha$. Prove that: $$\int_C \left(\frac {-y}{x^2+y^2}, \frac {x}{x^2+y^2}\right)ds=\varphi(b) - \varphi(a)$$
My first approach was to verify if the field was conservative, but I don't think that's the case. I have no clue on how to solve this. How can this be done?