Show that the integral of the form
$$\omega = \frac{xdy - ydx}{x^2 + y^2}$$
over the equation $$\frac{x^2}{a^2} + \frac{y^2}{b^2} =1$$
does not depend on a or b.
My solution:
$ d\omega = 0$ (tedious algebra)
Take $A = \{ \frac{x^2}{a^2} + \frac{y^2}{b^2} \leqslant1\}$ then $\partial A = \{ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1\}$
by Stokes $$\oint_{\partial A} \omega = \int_{A} d\omega = \int_{A}0 =0$$
Conclusion: Integral independent of choice for $a$ and $b$.
Now this is dubious to me because if a=b=1 then I would have $$\int_{A} d\theta = \int_{0}^{2\pi}d\theta =2\pi$$ - or I am missing something?
The error in your argument is that $\omega$ is not defined at then origin. To fix the error apply your argument to the region between the ellipse and the small circle $$x^2+y^2=\epsilon$$ for small enough $\epsilon$.