calculus , double integral at ellipse region Let S be the region satisfying $ 3x^2 + 2xy + y^2 \leq\ 1$ in the real plane $R^{2}$.
Then compute the value of the double integral 
$$\int\int_S\ e^{(3x^2+2xy+y^2)}dxdy$$
i learned double integral at polar coordinates.
but i didn't solve this problem.
Please give me hint or answer.

Referring to another answer 
and transforming the ellipse into:

$$(2+\sqrt{2})X^2+(2-\sqrt{2})Y^2=1$$

Let $(X,Y)=\displaystyle
\left(
  \frac{u\cos v}{\sqrt{2+\sqrt{2}}},
  \frac{u\sin v}{\sqrt{2-\sqrt{2}}}
\right)$, then $dX \, dY=\displaystyle \frac{u\, du \, dv}{\sqrt{2}}$
Now
$$\iint_{(2+\sqrt{2})X^2+(2-\sqrt{2})Y^2<1}
  e^{(2+\sqrt{2})X^2+(2-\sqrt{2})Y^2} dX \, dY
=\int_{0}^{2\pi} \int_{0}^{1} \frac{e^{u^2}}{\sqrt{2}} u\, du \,  dv$$
