Answer
$$4$$
Work Step by Step
$$\eqalign{
& \int_0^4 {\frac{{dx}}{{\sqrt {4 - x} }}} \cr
& {\text{The integrand is undefined for }}x = 4,{\text{ so the integral can be represented as}} \cr
& \int_0^4 {\frac{{dx}}{{\sqrt {4 - x} }}} = \mathop {\lim }\limits_{k \to {4^ - }} \int_0^k {\frac{{dx}}{{\sqrt {4 - x} }}} \cr
& {\text{Integrate}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{k \to {4^ - }} \left[ {2\sqrt {4 - x} } \right]_0^k \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{k \to {4^ - }} \left[ {2\sqrt {4 - k} - 2\sqrt {4 - 0} } \right] \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \mathop {\lim }\limits_{k \to {4^ - }} \left[ {2\sqrt {4 - k} - 4} \right] \cr
& \,{\text{Calculate the limit when }}k \to {4^ - } \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = - \left( {2\sqrt {4 - 4} - 4} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 4 \cr
& {\text{Then}}{\text{,}} \cr
& \int_0^4 {\frac{{dx}}{{\sqrt {4 - x} }}} = 4 \cr} $$
