Answer
$4\sqrt 3 - 6$
Work Step by Step
$$\eqalign{
& \text{Let } I=\int_3^4 {\sqrt {\frac{3}{x}} } dx \cr
& {\text{Using the radical property }}\sqrt {\frac{m}{n}} = \frac{{\sqrt m }}{{\sqrt n }} \cr
& I= \int_3^4 {\frac{{\sqrt 3 }}{{\sqrt x }}} dx \cr
& {\text{Using the radical property }}\sqrt x = {x^{ - 1/2}} \cr
& I = \int_3^4 {\frac{{\sqrt 3 }}{{{x^{1/2}}}}} dx \cr
& = \int_3^4 {\sqrt 3 } {x^{ - 1/2}}dx \cr
& = \sqrt 3 \int_3^4 {{x^{ - 1/2}}} dx \cr
& {\text{Integrate using the power rule}} \cr
& I = \sqrt 3 \left[ {\frac{{{x^{1/2}}}}{{1/2}}} \right]_3^4 \cr
& = 2\sqrt 3 \left[ {\sqrt x } \right]_3^4 \cr
& {\text{Using the fundamental theorem of calculus}}{\text{, part 2}} \cr
& I=2\sqrt 3 \left[ {\sqrt x } \right]_3^4 = 2\sqrt 3 \left( {\sqrt 4 - \sqrt 3 } \right) \cr
& {\text{Simplify}} \cr
& I=2\sqrt 3 \left( {2 - \sqrt 3 } \right) \cr
& = 4\sqrt 3 - 6 \cr} $$
