Calculate:
$\int\limits_R\!\!\int e^{-(x^2+xy+y^2)} dxdy $
where R is the region $ x^2 + y^2 + xy \le 1 $ .
I have tried to applied the polar coordinates followed by a rotation matrix, but the integrand kept looking like this $ e^{{\sin^2{\theta}}} $
An isometry maps $x^2+xy+y^2$ into the diagonalized quadratic form $\frac{3}{2}X^2+\frac{1}{2}Y^2$ and
$$\iint_{3X^2+Y^2\leq 2}\exp\left(-\frac{3}{2}X^2-\frac{1}{2}Y^2\right)\,dX\,dY =\frac{2}{\sqrt{3}}\iint_{u^2+v^2\leq 1}e^{-u^2-v^2}\,du\,dv$$ equals $$ \frac{4\pi}{\sqrt{3}}\int_{0}^{1} \rho e^{-\rho^2}\,d\rho = \frac{2\pi}{\sqrt{3}}(1-e^{-1}).$$
