Answer
$$ - \frac{{\sqrt {4 - {x^2}} }}{x} - {\sin ^{ - 1}}\frac{x}{2} + C\,\,\,$$
Work Step by Step
$$\eqalign{
& \int {\frac{{\sqrt {4 - {x^2}} }}{{{x^2}}}} dx \cr
& {\text{Use the Endpaper Integral Table to evaluate the integral}} \cr
& {\text{Rewrite the integrand}} \cr
& \int {\frac{{\sqrt {4 - {x^2}} }}{{{x^2}}}} dx = \int {\frac{{\sqrt {{2^2} - {x^2}} }}{{{x^2}}}} dx \cr
& {\text{The integrand has a expression in the form }}\sqrt {{a^2} - {u^2}} {} \cr
& {\text{Use formula 80}} \cr
& \left( {80} \right):\,\,\,\,\int {\frac{{\sqrt {{a^2} - {u^2}} }}{{{u^2}}}du} = - \frac{{\sqrt {{a^2} - {u^2}} }}{u} - {\sin ^{ - 1}}\frac{u}{a} + C\,\,\, \cr
& {\text{let }}u = x,\,\,\,a = 2 \cr
& \int {\frac{{\sqrt {{2^2} - {x^2}} }}{{{x^2}}}} dx = - \frac{{\sqrt {{2^2} - {x^2}} }}{x} - {\sin ^{ - 1}}\frac{x}{2} + C\,\,\, \cr
& {\text{simplifying}} \cr
& \int {\frac{{\sqrt {4 - {x^2}} }}{{{x^2}}}} dx = - \frac{{\sqrt {4 - {x^2}} }}{x} - {\sin ^{ - 1}}\frac{x}{2} + C\, \cr} $$
