Answer
$${\text{Choose }}u = {\tan ^{ - 1}}x{\text{ and }}dv = dx$$
Work Step by Step
$$\eqalign{
& \int {{{\tan }^{ - 1}}x} dx \cr
& {\text{Choose }}u = {\tan ^{ - 1}}x{\text{ and }}dv = dx.\,\,\,{\text{Then,}} \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,du = \frac{1}{{1 + {x^2}}}dx,\,\,\,\,\,\,v = x \cr
& {\text{Using the integration by parts formula}} \cr
& \int {udv} = uv - \int {vdu} \cr
& \int {{{\tan }^{ - 1}}xdx} = x{\tan ^{ - 1}}x - \int {\frac{x}{{1 + {x^2}}}d} x \cr
& \int {{{\tan }^{ - 1}}xdx} = x{\tan ^{ - 1}}x - \frac{1}{2}\ln \left( {1 + {x^2}} \right) + C \cr} $$
