Answer
$$\frac{x}{{3\sqrt {{x^2} + 3} }} + C$$
Work Step by Step
$$\eqalign{
& \int {\frac{{dx}}{{{{\left( {3 + {x^2}} \right)}^{3/2}}}}} \cr
& {\text{Use the Endpaper Integral Table to evaluate the integral }} \cr
& {\text{For integrals containing }}{\left( {{a^2} + {u^2}} \right)^{3/2}}{\text{, use the formula 98}} \cr
& \left( {98} \right):\int {\frac{{du}}{{{{\left( {{u^2} + {a^2}} \right)}^{3/2}}}}} = \frac{u}{{{a^2}\sqrt {{u^2} + {a^2}} }} + C \cr
& \cr
& \int {\frac{{dx}}{{{{\left( {3 + {x^2}} \right)}^{3/2}}}}} = \int {\frac{{dx}}{{{{\left( {{{\left( {\sqrt 3 } \right)}^2} + {x^2}} \right)}^{3/2}}}}} \cr
& {\text{Then}}{\text{, let }}a = \sqrt 3 \cr
& \int {\frac{{dx}}{{{{\left( {{{\left( {\sqrt 3 } \right)}^2} + {x^2}} \right)}^{3/2}}}}} = \frac{x}{{{{\left( {\sqrt 3 } \right)}^2}\sqrt {{x^2} + {{\left( {\sqrt 3 } \right)}^2}} }} + C \cr
& {\text{simplifying}} \cr
& \int {\frac{{dx}}{{{{\left( {3 + {x^2}} \right)}^{3/2}}}}} = \frac{x}{{3\sqrt {{x^2} + 3} }} + C \cr} $$
