Question

Evaluate the integral $\int_{0}^{\infty} \int_{0}^{\infty} \frac{dx\,dy}{(1+x^2+y^2)^2}$

Answer 1

$\frac{\pi}{4}$

Answer 2

The integral is transformed into polar coordinates, where $x=r\cos\theta$, $y=r\sin\theta$, and $dx\,dy=r\,dr\,d\theta$. The first quadrant ($x\ge0, y\ge0$) corresponds to $r\in[0,\infty)$ and $\theta\in[0,\pi/2]$. The integral becomes $\int_{0}^{\pi/2} \int_{0}^{\infty} \frac{r\,dr\,d\theta}{(1+r^2)^2}$. This is separated into an angular integral ($\int_{0}^{\pi/2} d\theta = \pi/2$) and a radial integral ($\int_{0}^{\infty} \frac{r\,dr}{(1+r^2)^2}$). Using the substitution $u=1+r^2$ for the radial integral, we get $\frac{1}{2}\int_{1}^{\infty} u^{-2}du = \frac{1}{2}$. The final result is the product of these two values: $(\pi/2) \times (1/2) = \pi/4$.