Answer
$2$
Work Step by Step
$x$ = $tanθ$
$dx$ = $sec^{2}θdθ$
$\int{\frac{dx}{(1+x^{2})^{\frac{3}{2}}}}$ = $\int{\frac{sec^{2}θ}{sec^{3}θ}}dθ$ = $\int{cosθdθ}$ = $sinθ+C$ = $\frac{x}{\sqrt {1+x^{2}}}+C$
$\int_0^{\infty}{\frac{dx}{(1+x^{2})^{\frac{3}{2}}}}$ = $\lim\limits_{R \to \infty}$$\int_0^{R}{\frac{dx}{(1+x^{2})^{\frac{3}{2}}}}$ = $\lim\limits_{R \to \infty}$$\frac{R}{\sqrt {1+R^{2}}}$ = $1$
$\int_{-\infty}^{0}{\frac{dx}{(1+x^{2})^{\frac{3}{2}}}}$ = $\lim\limits_{R \to -\infty}$$\int_R^{0}{\frac{dx}{(1+x^{2})^{\frac{3}{2}}}}$ = $\lim\limits_{R \to -\infty}$$-\frac{R}{\sqrt {1+R^{2}}}$ = $1$
$\int_{-\infty}^{-\infty}{\frac{dx}{(1+x^{2})^{\frac{3}{2}}}}$ = $1+1$ = $2$
