Answer
$\dfrac{\pi}{4}$
Work Step by Step
Our aim is to integrate the integral as follows:
$\int^{\infty}_0 \int^{\infty}_0 \dfrac{1}{(1+x^2+y^2)^2} \space dx \space dy =\int^{\pi/2}_0 \int^{\infty}_0 \dfrac{r}{(1+r^2)^2} \space dr \space d\theta $
or, $=\dfrac{\pi}{2} \lim\limits_{b \to \infty} \int^b_0 \dfrac{r}{(1+r^2)^2} \space dr $
or, $=\dfrac{\pi}{4} \lim\limits_{b \to \infty}[1-\dfrac{1}{1+r^2}]_0^b $
or, $=\dfrac{\pi}{4} \times \lim\limits_{b \to \infty}(1-\dfrac{1}{1+b^2})$
or, $=\dfrac{\pi}{4}$
