Answer
$$\frac{1}{{\sqrt {{a^2} - {u^2}} }}$$
Work Step by Step
$$\eqalign{
& \int {\frac{{du}}{{\sqrt {{a^2} - {u^2}} }}} = \arcsin \frac{u}{a} + C \cr
& {\text{Differentiating}} \cr
& \frac{d}{{dx}}\left[ {\arcsin \frac{u}{a} + C} \right] = \frac{{\frac{d}{{dx}}\left[ {\frac{u}{a}} \right]}}{{\sqrt {1 - {{\left( {\frac{u}{a}} \right)}^2}} }} + \frac{d}{{dx}}\left[ C \right] \cr
& = \frac{{\frac{1}{a}}}{{\sqrt {1 - \frac{{{u^2}}}{{{a^2}}}} }} + 0 \cr
& {\text{Simplifying}} \cr
& = \frac{{\frac{1}{a}}}{{\sqrt {\frac{{{a^2} - {u^2}}}{{{a^2}}}} }} + 0 \cr
& = \frac{1}{{\sqrt {{a^2} - {u^2}} }} \cr} $$
