5.5 The Substitution Rule/43

Failed to parse (Conversion error. Server ("https://en.wikipedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle {\begin{aligned}\int {\frac {1+x}{1+x^{2}}}dx&=\int ({\frac {1}{1+x^{2}}}+{\frac {x}{1+x^{2}}})dx\\[2ex]&=\int {\frac {1}{1+x^{2}}}dx+\int {\frac {x}{1+x^{2}}}dx\\[2ex]&=tan^{-1}(x)+{\frac {1}{2}}\int {\frac {1}{u}}du\\[2ex]&=tan^{-1}(x)+\ln |{1+x}|+c\end{aligned}}}