x[t_] = 6 Cos[t];
y[t_] = 6 Sin[t];
m[x_, y_] = (1 - y)/((x + 2)^2 + (y - 1)^2);
n[x_, y_] = (x + 2)/((x + 2)^2 + (y - 1)^2);
Together[D[n[x, y], x] - D[m[x, y], y]]
Integrate[m[x[t], y[t]] * x'[t] + n[x[t], y[t]] * y'[t], {t, 0, 2Pi}]