Consider the $1$-form $$\alpha=\frac{1}{x^2+y^2}(-ydx+xdy)\in \Omega^1(\mathbb{R}^2\setminus \{(0,0)\})$$ If $\gamma:[0,1]\to \mathbb{R}^2\setminus \{(0,0)\}$ is such that $\gamma(t)=(x(t),y(t))$ where $y(t)>0$ for all $t\in [0,1]$ and $\gamma(0)=\gamma(1)$, find $\int_{\gamma}\alpha$.
I've been stuck on this for a while and am not sure how to get started. Going through this post, the two answers proposed, to be honest, are confusing me (which is mainly because I'm quite new to differential forms). Can someone please explain, without using Stokes' theorem (since this has not been covered yet in my class) or contraction, how I am supposed to use the assumption that $y(t)>0$ for all $t\in [0,1]$ to evaluate the integral?