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Question Number 146455 by physicstutes last updated on 13/Jul/21
Given that  F(x) = ∫_0 ^x (t^2 /( (√(t^2 +1))))dt    Show that F(x) is an increasing function
$$\mathrm{Given}\:\mathrm{that}\:\:{F}\left({x}\right)\:=\:\int_{\mathrm{0}} ^{{x}} \frac{{t}^{\mathrm{2}} }{\:\sqrt{{t}^{\mathrm{2}} +\mathrm{1}}}{dt}\: \\ $$$$\:\mathrm{Show}\:\mathrm{that}\:{F}\left({x}\right)\:\mathrm{is}\:\mathrm{an}\:\mathrm{increasing}\:\mathrm{function} \\ $$
Answered by gsk2684 last updated on 13/Jul/21
F^  (x)=(x^2 /( (√(x^2 +1))))≥0  F(x) is an increasing function
$${F}^{ } \left({x}\right)=\frac{{x}^{\mathrm{2}} }{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}\geqslant\mathrm{0} \\ $$$${F}\left({x}\right)\:{is}\:{an}\:{increasing}\:{function} \\ $$

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