Convergence of an improper integral: coordinate change

Question

I have to study the convergence of the following integral

$$ \iint_D \frac{1}{x^2+(y-1)^2} dxdy $$

on the domain $D=\{(x,y): x+y \ge 1\}$.

My problem concerns the coordinate change. I put $u=x+y$ and $v=y$, so $u \in (1,+\infty)$ and $v \in \mathbb{R}$. But this is impossible, since when $v \to -\infty$, $u$ must tend to the same limit, contradicting my choice. How can I solve the problem?

Answer 1

That is just $$\int_{-\infty}^{+\infty}\int_{1-x}^{+\infty}\frac{1}{x^2+(1-y)^2}\,dy\,dx =\int_{-\infty}^{+\infty}\frac{\pi+2\pi\,\text{sign}(x)}{4x}\,dx$$ that is diverging.

Answer 2

Hint: see the drawing:

Dashed: $x + y= $ constant. Dotted: interval of variation of $y$.