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Question Number 146697 by ArielVyny last updated on 15/Jul/21
∀t≥−1,F(t)=(2/π)∫_0 ^(π/2) (√(1+tcos^2 ϕ))dϕ  1) Show that ∀t≤−1 F(t)=(√(1+t))F(−(1/(1+t)))  2) show that if 0≤t_1  ,  0≤F(t_2 )−F(t_1 )≤((t_2 −t_1 )/4)
$$\forall{t}\geqslant−\mathrm{1},{F}\left({t}\right)=\frac{\mathrm{2}}{\pi}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \sqrt{\mathrm{1}+{tcos}^{\mathrm{2}} \varphi}{d}\varphi \\ $$$$\left.\mathrm{1}\right)\:{Show}\:{that}\:\forall{t}\leqslant−\mathrm{1}\:{F}\left({t}\right)=\sqrt{\mathrm{1}+{t}}{F}\left(−\frac{\mathrm{1}}{\mathrm{1}+{t}}\right) \\ $$$$\left.\mathrm{2}\right)\:{show}\:{that}\:{if}\:\mathrm{0}\leqslant{t}_{\mathrm{1}} \:, \\ $$$$\mathrm{0}\leqslant{F}\left({t}_{\mathrm{2}} \right)−{F}\left({t}_{\mathrm{1}} \right)\leqslant\frac{{t}_{\mathrm{2}} −{t}_{\mathrm{1}} }{\mathrm{4}} \\ $$

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