∀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


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Question Number 146697 by ArielVyny last updated on 15/Jul/21

$\forall t ⩾ - 1, F (t) = \frac{2}{π} \int_{0}^{\frac{π}{2}} \sqrt{1 + {t c o s}^{2} φ} d φ$ $1) S h o w t h a t \forall t ⩽ - 1 F (t) = \sqrt{1 + t} F (- \frac{1}{1 + t})$ $2) s h o w t h a t i f 0 ⩽ t_{1},$ $0 ⩽ F (t_{2}) - F (t_{1}) ⩽ \frac{t_{2} - t_{1}}{4}$
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