Answer
$$5{\sin ^{ - 1}}\left( {\frac{x}{5}} \right) + \sqrt {25 - {x^2}} + C$$
Work Step by Step
$$\eqalign{
& \int {\sqrt {\frac{{5 - x}}{{5 + x}}} dx} \cr
& {\text{Recall that }}\sqrt {\frac{a}{b}} = \frac{{\sqrt a }}{{\sqrt b }} \cr
& = \int {\frac{{\sqrt {5 - x} }}{{\sqrt {5 + x} }}dx} \cr
& {\text{Rationalizing the numerator}} \cr
& = \int {\frac{{\sqrt {5 - x} }}{{\sqrt {5 + x} }} \times \frac{{\sqrt {5 - x} }}{{\sqrt {5 - x} }}dx} \cr
& = \int {\frac{{{{\left( {\sqrt {5 - x} } \right)}^2}}}{{\sqrt {{{\left( 5 \right)}^2} - {{\left( x \right)}^2}} }}dx} \cr
& = \int {\frac{{5 - x}}{{\sqrt {25 - {x^2}} }}dx} \cr
& {\text{Distribute}} \cr
& = \int {\frac{5}{{\sqrt {25 - {x^2}} }}dx} - \int {\frac{x}{{\sqrt {25 - {x^2}} }}dx} \cr
& {\text{Rewrite}} \cr
& = 5\int {\frac{1}{{\sqrt {25 - {x^2}} }}dx} + \frac{1}{2}\int {\frac{{ - 2x}}{{\sqrt {25 - {x^2}} }}dx} \cr
& {\text{Integrate by basic formulas of integration}} \cr
& = 5{\sin ^{ - 1}}\left( {\frac{x}{5}} \right) + \sqrt {25 - {x^2}} + C \cr} $$
