I have the follwing problem:
Let $$f_a(x,y)=\frac{1}{\left(\left(x+y-1\right)^2+\left(x-y+2\right)^2+1\right)^a}$$ For which $a>0$ is the function integrable.
I have just shown that $f_a$ is mesurable. Now I only need to compute $$\int_\mathbb{R}\left(\int_\mathbb{R} |f_a(x,y)|d\lambda(x)\right)d\lambda(y)$$ To do so, they give us the hint to use a change of variables and then the polar coordinates. But I somehow don't see how to do the change of variables. I thought that maybe one could first apply the polar coordinates but then I get something like $$\left|\frac{1}{\left(2r^2+2r(\cos\theta-3\sin\theta)+6\right)^a}\right|$$ Could maybe someone give me a hint?
Hint: The change of coordinates referred to was actually
$$\begin{cases}u = x+y-1\\v=x-y+2\end{cases} \implies J^{-1} = 2$$
which would give an integrand of
$$\iint_{\Bbb{R}^2} \frac{dudv}{2(u^2+v^2+1)^a}$$
and then change to polar coordinates after to analyze convergence.