Computing Double Integrals Using Green's Theorem

Question

Consider the region $A = \{(x,y) \in \mathbb{R}^2 : 1/2 \leq \sqrt{x^2+y^2} \leq 2\}$, i.e, an Annulus with inner radius $1/2$, outer radius $2$. I would like to compute: $$ \iint_A 2x - 2ye^{x^2+y^2} \mathrm{d}x\mathrm{d}y $$ I would like to compute this using Green's theorem, i.e., rewriting the above integral as a line-integral. The default statement of Green's theorem motivates us to set: $$ \partial_xQ = 2x $$ $$ \partial_yP = -2ye^{x^2 + y^2} $$ Thus, we have that: $$ Q = x^2 + C(y) $$ $$ P = e^{x^2 + y^2} + C(x) $$ after integrating the derivatives with respect to $x,y$, respectively. Given, this, how should I go about computing: $$ \oint_{\partial A} P \mathrm{d}x + Q \mathrm{d}y $$ ?