Integration of (1+x)/(1+x^2)

 

\begin{array}{l} \smallint \frac{{1 + x}}{{1 + {x^2}}}{\rm{d}}x\\\\ \smallint \frac{1}{{{x^2} + 1}}dx + \smallint \frac{x}{{{x^2} + 1}}dx\\ By\;substitution\\ let \Rightarrow u = {x^2} + 1\\ du = 2xdx\\ {\tan ^{ - 1}}x + \frac{1}{2}\smallint \frac{1}{u}du\\ {\tan ^{ - 1}}x + \frac{1}{2}\ln u + c\\ \Rightarrow {\tan ^{ - 1}}x + \frac{1}{2}\ln \left( {{x^2} + 1} \right) + c \end{array}