<s>[INST] How to evaluate this double integral over a semicircle? $$\iint_{D}\frac{x^4+y^4}{1+e^{3x^2y-y^3}} dxdy$$
$D = \{(x,y):x^2+y^2 \leq 1,x>0\}$. From the shape of region $D$ it seems to me that it's better to convert it to polar coordinate,but with no luck.
 [/INST] Hint 1:
$$x^4 + y^4 = r^4 - 2x^2y^2 = r^4-2r^4\cos^2(\theta)\sin^2(\theta)=\frac{1}{2}r^4\sin^2(2\theta)$$
Hint 2:
$$3x^2y - y^3 = y(3x^2 - y^2)= r^3\sin(\theta)\left(3\cos^2(\theta) - \sin^2(\theta)\right)$$
$$=2 \cos(2 \theta)+1$$
</s>