Find $ \oint_c \frac{-y}{x^2+4y^2}dx+\frac{x}{x^2+4y^2}dy$ when $c$ is the unit circle.

Question

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 ?