I just wanted to ask if the provided solution to the following integral is correct.
Prove that $$ \lim_{r\to+\infty}\int_{B(0,r)}\frac{1}{1+x^2+|y|}dx\,dy=\infty $$
Solution
The above integral equals
$$ \int_{\mathbb{R}^2}\frac{1}{1+x^2+|y|}dx\,dy $$
Integrating first in $x$ and then in $y$, one gets
$$ \begin{align} \int_{\mathbb{R}^2}\frac{1}{1+x^2+|y|}dx\,dy &= 4\int_0^{+\infty}\int_0^{+\infty}\frac{1}{1+x^2+y}dx\,dy \\[8pt] &=2\pi\int_0^{+\infty}\frac{1}{\sqrt{1+y}}dy \\[4pt] &=\infty \end{align}$$
More specifically, I wanted to ask if transforming the limit in the exercise into the integral over the whole $\mathbb{R}^2$ is correct.