Find $$ \oint_c \frac{-y}{x^2+4y^2}dx+\frac{x}{x^2+4y^2}dy$$ when $c$ is the unit circle (Counterclockwise).
My attempt:
Denote $P=\frac{-y}{x^2+4y^2}, Q=\frac{x}{x^2+4y^2}$.
$Q_x=\frac{4y^2-x^2}{(x^2+4y^2)^2}, P_y=\frac{4y^2-x^2}{(x^2+4y^2)^2}$
Using Green's Theorem $ \oint_c \frac{-y}{x^2+4y^2}dx+\frac{x}{x^2+4y^2}dy = \int\int_c Q_x-P_y=\int\int_c 0=0$
I am not entirely sure that I can use Green's Theorem , is my solution correct ?