In cylindrical coordinates, we take
[tex]x=r\cos\theta[/tex]
[tex]y=r\sin\theta[/tex]
[tex]z=z[/tex]
so that [tex]\mathrm dx\,\mathrm dy\,\mathrm dz=r\,\mathrm dr\,\mathrm d\theta\,\mathrm dz[/tex].
We have
[tex]1+x^2+y^2=1+r^2[/tex]
and the integral is
[tex]\displaystyle\int_0^2\int_0^\pi\int_0^1\frac r{1+r^2}\,\mathrm dr\,\mathrm d\theta\,\mathrm dz=\frac{\ln2}2\int_0^2\int_0^\pi\mathrm d\theta\,\mathrm dz=\boxed{\pi\ln2}[/tex]
