Example Evaluate the following improper integral．

$$\iint_{\Omega}\frac{dxdy}{1 + (x^2 + y^2)^2}, \ \Omega = \{(x,y) : -\infty < x,y < \infty \}$$

Answer $\Omega = \{(x,y) : -\infty < x, y < \infty\}$ represents whole $xy$plane. So, it is not bounded. So, we consider the following region.

$$\Omega_{n} = \{(x,y) : x^2 + y^2 \leq n^2\} \ (n \geq 1)$$

Thne $\Omega = \lim_{n \to \infty}\Omega_{n}$．Now we apply polar coordinate. Then

$$\Gamma_{n} = \{(r,\theta) : 0 \leq \theta \leq 2\pi, 0 \leq r \leq n\}$$

Thus,

$$I_{n} = \iint_{\Omega_{n}}\frac{1}{1 + (x^2 + y^2)^2}dxdy = \iint_{\Gamma_{n}}\frac{1}{1 + r^4}\vert J\vert dr d\theta$$

$$= \int_{0}^{2\pi}\int_{0}^{n}\frac{r}{1 + r^4}dr d\theta \ \left(\b... ...rac{1}{2}\tan^{-1}{t} + c\\ = \frac{1}{2}\tan^{-1}{r^2} + c \end{array}\right)$$

$$= \int_{0}^{2\pi} d\theta\left[\frac{1}{2}\tan^{-1}{r^2}\right]_{0}^{2\pi}$$

$$= 2\pi \cdot \frac{1}{2}(\tan^{-1}n^2 - \tan^{-1}{0})$$

Therefore,

$$I = \iint_{\Omega}\frac{1}{1 + (x^2 + y^2)^2}dxdy = \lim_{n \to \infty}I_{n} = \pi \cdot \frac{\pi}{2} = \frac{\pi^2}{2}.$$