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Author: Tinku Tara

Evaluate-0-1-tan-1-x-x-dx-

Question Number 12101 by Nayon last updated on 13/Apr/17 $${Evaluate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{tan}^{−\mathrm{1}} \left({x}\right)}{\:{x}}{dx} \\ $$ Answered by b.e.h.i.8.3.4.1.7@gmail.com last updated on 13/Apr/17 $${x}={tg}\varphi\Rightarrow{dx}=\frac{{d}\varphi}{\mathrm{1}+{tg}^{\mathrm{2}} \varphi}\begin{pmatrix}{{x}=\mathrm{1}\Rightarrow\varphi=\frac{\pi}{\mathrm{4}}}\\{{x}=\mathrm{0}\Rightarrow\varphi=\mathrm{0}}\end{pmatrix} \\…

arctan-x-1-dx-propose-par-Rodrigue-

Question Number 143167 by pticantor last updated on 10/Jun/21 $$\int\boldsymbol{{arctan}}\left(\sqrt{\sqrt{\boldsymbol{{x}}}+\mathrm{1}}\right)\boldsymbol{{dx}}=??? \\ $$$$\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{propose}}'\:\boldsymbol{{par}}\:\boldsymbol{{Rodrigue}} \\ $$ Answered by Olaf_Thorendsen last updated on 10/Jun/21 $$\mathrm{F}\left({x}\right)\:=\:\int\mathrm{arctan}\left(\sqrt{\sqrt{{x}}+\mathrm{1}}\right){dx} \\…