Answer
$\sqrt {{x^2} - 7} + C$
Work Step by Step
$$\eqalign{
& \int {\frac{x}{{\sqrt {{x^2} - 7} }}} dx \cr
& {\text{Integrate by the substitution method}} \cr
& {\text{Let }}u = {x^2} - 7,{\text{ }}du = 2xdx,{\text{ }}xdx = \frac{1}{2}du \cr
& {\text{Substituting}} \cr
& \int {\frac{x}{{\sqrt {{x^2} - 7} }}} dx = \int {\frac{{\left( {1/2} \right)}}{{\sqrt u }}} du \cr
& = \frac{1}{2}\int {\frac{1}{{\sqrt u }}} du \cr
& = \frac{1}{2}\int {{u^{ - 1/2}}} du \cr
& {\text{Use the power rule }}\int {{u^n}} du = \frac{{{u^{n + 1}}}}{{n + 1}} + C \cr
& = \frac{1}{2}\left( {\frac{{{u^{1/2}}}}{{1/2}}} \right) + C \cr
& = \sqrt u + C \cr
& {\text{Write in terms of }}x,{\text{ substitute }}{x^2} - 7{\text{ for }}u \cr
& = \sqrt {{x^2} - 7} + C \cr} $$
