I'm trying to solve the following exercise:
Let $\omega = \frac{xdy-ydx}{x^2+y^2}$ on $\mathbb{R}^2 \setminus (0,0)$ (which is the standard example of a closed but not exact form). Let $g\colon[0,2\pi] \to \mathbb{R}^2 \setminus (0,0)$ be defined by $g(t)=(e^t \sin(17t),e^{t^2} \cos(17t))$ Calculate the integral of $g^*(\omega)$ over $[0, 2\pi]$
Just by looking at this question, I think it's not good idea to actually calculate (a tried... :) ) this. Which means this thing has to be $0$ or $ 2 \pi$. I'm tending towards $2 \pi$ since although $g$ is not a closed curve, we're still somehow doing a full rotation around (0,0).
But I just have no idea where to start and or what I have to show. Could someone give me hint on how to solve such a problem?
Thanks for any help.
Edit, here's g: