Answer
$\pi^2$
Work Step by Step
$$I = \int_{-\infty}^{\infty} \dfrac{dx}{x^2+1} \int_{-\infty}^{\infty} \dfrac{dy}{y^2+1} \\=[ \tan^{-1} x]_{-\infty}^{\infty} [ \tan^{-1} y]_{-\infty}^{\infty} \\=[\tan^{-1} x (-\infty) -\tan^{-1} x (\infty)] [\tan^{-1} y (-\infty) -\tan^{-1} y (\infty)] \\=(\pi)( \pi) \\=\pi^2$$
