$C$ is a closed curve, $(0,0)$ is surrounded by $C$, Let $X = ax+by,\quad Y= cx+dy,\quad ad-bc=-7$, compute integral $$ \int_C \frac{X \; dY-Y \; dX}{X^2+Y^2} $$
One of my idea is $$d(\arctan\frac{X}{Y})=\frac{X \; dY-Y \; dX}{X^2+Y^2}$$ then $ \int_C \frac{X \; dY-Y \; dX}{X^2+Y^2} = \int_Cd(\arctan\frac{X}{Y})$, But I can't draw more conclusions.
I also want to use Green's formula, if $X=x$, then I got that integral is zero. but $X\neq x$.
And, what is the use of the hypothesis $ad-bc=-7$?
Thanks for your help.