Answer
$$ - \frac{{\sqrt {{x^2} - 5} }}{x} + \ln \left| {x + \sqrt {{x^2} - 5} } \right| + C\,\,\,$$
Work Step by Step
$$\eqalign{
& \int {\frac{{\sqrt {{x^2} - 5} }}{{{x^2}}}} dx \cr
& {\text{Use the Endpaper Integral Table to evaluate the integral}} \cr
& {\text{Rewrite the integrand}} \cr
& \int {\frac{{\sqrt {{x^2} - 5} }}{{{x^2}}}} dx = \int {\frac{{\sqrt {{x^2} - {{\left( {\sqrt 5 } \right)}^2}} }}{{{x^2}}}} dx \cr
& {\text{The integrand has a expression in the form }}\sqrt {{u^2} - {a^2}} {} \cr
& {\text{Use formula 94}} \cr
& \left( {94} \right):\,\,\,\,\int {\frac{{\sqrt {{u^2} - {a^2}} }}{{{u^2}}}du} = - \frac{{\sqrt {{u^2} - {a^2}} }}{u} + \ln \left| {u + \sqrt {{u^2} - {a^2}} } \right| + C\,\,\, \cr
& {\text{let }}u = x,\,\,\,a = \sqrt 5 \cr
& \int {\frac{{\sqrt {{x^2} - {{\left( {\sqrt 5 } \right)}^2}} }}{{{x^2}}}} dx = - \frac{{\sqrt {{x^2} - {{\left( {\sqrt 5 } \right)}^2}} }}{x} + \ln \left| {x + \sqrt {{x^2} - {{\left( {\sqrt 5 } \right)}^2}} } \right| + C\,\,\, \cr
& {\text{simplifying}} \cr
& \int {\frac{{\sqrt {{x^2} - 5} }}{{{x^2}}}} dxdx = - \frac{{\sqrt {{x^2} - 5} }}{x} + \ln \left| {x + \sqrt {{x^2} - 5} } \right| + C\,\,\, \cr} $$
