Answer
$$8$$
Work Step by Step
$$\eqalign{
& \int_{ - 2}^6 {\frac{{dx}}{{\sqrt {\left| {x - 2} \right|} }}} \cr
& {\text{The integrand is not continuous at }}x = 2,{\text{ then}} \cr
& \int_{ - 2}^6 {\frac{{dx}}{{\sqrt {\left| {x - 2} \right|} }}} = \mathop {\lim }\limits_{a \to {2^ - }} \int_{ - 2}^a {\frac{{dx}}{{\sqrt {\left| {x - 2} \right|} }}} + \mathop {\lim }\limits_{b \to {2^ + }} \int_b^6 {\frac{{dx}}{{\sqrt {\left| {x - 2} \right|} }}} \cr
& {\text{Integrating}} \cr
& = - 2\mathop {\lim }\limits_{a \to {2^ - }} \left[ {\sqrt {2 - x} } \right]_{ - 2}^a + 2\mathop {\lim }\limits_{b \to {2^ + }} \left[ {\sqrt {x - 2} } \right]_b^6 \cr
& = - 2\mathop {\lim }\limits_{a \to {2^ - }} \left[ {\sqrt {2 - a} - \sqrt 4 } \right] + 2\mathop {\lim }\limits_{b \to {2^ + }} \left[ {\sqrt {6 - 2} - \sqrt {b - 2} } \right] \cr
& = - 2\mathop {\lim }\limits_{a \to {2^ - }} \left[ {\sqrt {2 - a} - 2} \right] + 2\mathop {\lim }\limits_{b \to {2^ + }} \left[ {2 - \sqrt {b - 2} } \right] \cr
& {\text{Evaluating the limit}} \cr
& = - 2\left[ {\sqrt {2 - 2} - 2} \right] + 2\left[ {2 - \sqrt {2 - 2} } \right] \cr
& = 4 + 4 \cr
& = 8 \cr} $$
