Finding a variable substitution for a double integral I have the following double integral
$$
\iint_D xdxdy
$$
where D is given by the inequalities
$$
x^2+xy+y^2 \le 4, x\ge 0
$$
In the solution given to the problem they apply this variable substitution when solving the integral:
$$
u = \frac{\sqrt{3}}{2}x
$$
$$
v = \frac{x}2 + y
$$
My question is how they arrive at this substitution? Is there a method for finding substitutions like these?
While I understand that this substitution works, there is no explanation for how they arrive at this substitution at all in my solution.

You can transform the region D by completing the square
$$x^2+xy+y^2=\frac{3}{4}x^2+\frac{1}{4}x^2+xy+y^2=(\frac{1}{2}x+y)^2+(\frac{\sqrt{3}}{2}x)^2\leq 4$$
Let $u = \frac{\sqrt{3}}{2}x, v=\frac{x}{2}+y$. And notice that $x\geq 0$ gives $u\geq 0$ too.
