<s>[INST] Improper double integrals $$
\iint_{\mathbb{R}^{2}}{\ln\left(1 + 2x^{2} + y^{2}\right) \over 1+x^{2}+y^{2}}\,\mathrm{d}x\,\mathrm{d}y
$$ 
using
$\ln\left(1 + 2x^{2} + y^{2}\right) \leq
\,\sqrt{\, 1 + x^{2} + y^{2}\,}\,$ and showed convergence by the comparison test, but according to my friends it diverges. 
Who is right ?. Thanks in advance.
 [/INST] The integral diverges as $x^2+y^2\to \infty$.  To see this, we note that
$$\begin{align}
\iint_{1\le x^2+y^2\le R}\frac{\log(1+2x^2+y^2)}{x^2+y^2+1}\,dx\,dy &\ge \frac{\log(2)}{2}\iint_{1\le x^2+y^2\le R}\frac{1}{x^2+y^2}\,dx\,dy\\\\
&=\frac{\log(2)}{2}\int_0^{2\pi}\int_1^R\frac{1}{r}\,dr\,d\theta\\\\
&=\pi \log(2R)\\\\
&\to \infty \,\,\text{as}\,\,R\to \infty
\end{align}$$
</s>