Question Number 146697 by ArielVyny last updated on 15/Jul/21
$$\forall{t}\geqslant−\mathrm{1},{F}\left({t}\right)=\frac{\mathrm{2}}{\pi}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{1}+{tcos}^{\mathrm{2}} \varphi}{d}\varphi \\ $$$$\left.\mathrm{1}\right)\:{Show}\:{that}\:\forall{t}\leqslant−\mathrm{1}\:{F}\left({t}\right)=\sqrt{\mathrm{1}+{t}}{F}\left(−\frac{\mathrm{1}}{\mathrm{1}+{t}}\right) \\ $$$$\left.\mathrm{2}\right)\:{show}\:{that}\:{if}\:\mathrm{0}\leqslant{t}_{\mathrm{1}} \:, \\ $$$$\mathrm{0}\leqslant{F}\left({t}_{\mathrm{2}} \right)−{F}\left({t}_{\mathrm{1}} \right)\leqslant\frac{{t}_{\mathrm{2}} −{t}_{\mathrm{1}} }{\mathrm{4}} \\ $$