I am self-learning integration on manifolds, and I'm trying to find an answer to the following question.
For the manifold $M=\{(x,y) \in \mathbb{R} : (x,y) \neq (0,0) \}$, let $f: M \rightarrow \mathbb{R}$ be a smooth function and $\omega$ a smooth 1-form on $M$ given by
$\omega = f(xdx + ydy)$.
Let $C: [0,1] \rightarrow M$ be the curve given by
$C(t) = (cos (2\pi t), sin (2\pi t))$ for $0 \leq t \leq 1$.
I am trying to compute $\int_{C} \omega$ and then show that $dw=0$ if and only if $f=F(x^2+y^2)$ for some $F$.
I know that one would integrate a general $\omega = p(x,y)dx + q(x,y)dy$ along a curve C as follows:
$\int_{C} p(x,y)dx + q(x,y)dy = \int_0^1 (p(C(t))\frac{dC^x}{dt}+q(C(t))\frac{dC^y}{dt})$, where the latter is the pullback $C^\star \omega$.
However, I am still fairly new to this and I do not know how to approach this given this generic smooth function $f$. How is this done?