Question Number 167150 by mathlove last updated on 08/Mar/22
Answered by som(math1967) last updated on 08/Mar/22
$$\:{let}\:\left({arctanx}^{{arctanx}^{…} } \right)={z} \\ $$$$\:{arctanx}^{{arctanx}^{…} } {lnarctanx}={lnz} \\ $$$$\:{zlnarctanx}={lnz} \\ $$$$\:\frac{{dz}}{{dx}}{lnarctanx}+\frac{{z}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right){arctanx}}=\frac{\mathrm{1}}{{z}}\frac{{dz}}{{dx}} \\ $$$$\:\Rightarrow\:{dz}\:=\frac{{z}^{\mathrm{2}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right){arctanx}}\:×\frac{\mathrm{1}}{\left(\mathrm{1}−{zlnarctanx}\right)}{dx} \\ $$$$=\frac{\left({arctanx}^{{arctanx}^{…} } \right)^{\mathrm{2}} {dx}}{{arctanx}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}−{arctanx}^{{arctanx}^{..} } {lnarctanx}\right)} \\ $$$$\int{dz}={z}+{c} \\ $$$$={arctanx}^{{arctanx}^{…} } +{c} \\ $$