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