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Category: Coordinate Geometry

Question-198434

Question Number 198434 by cortano12 last updated on 20/Oct/23 Commented by mr W last updated on 20/Oct/23 $${due}\:{to}\:{symmetry} \\ $$$${max}=\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)=\frac{\sqrt{\mathrm{2}}}{\mathrm{4}} \\ $$ Answered by mr…

Soit-f-x-2x-1-lnx-ln-1-x-et-I-1-0-f-x-dx-1-Montrer-que-n-2-1-4n-I-pi-2n-3-k-1-n-1-k-n-k-sin-kpi-n-3-4n-2-En-de-duire-la-valeur-de-I-

Question Number 198145 by Erico last updated on 11/Oct/23 $$\mathrm{Soit}\:\mathrm{f}\left(\mathrm{x}\right)=\frac{\mathrm{2x}−\mathrm{1}}{\mathrm{lnx}−\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)}\:\mathrm{et}\:\mathrm{I}=\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\ $$$$\mathrm{1}.\:\mathrm{Montrer}\:\mathrm{que}\:\forall\mathrm{n}\geqslant\mathrm{2} \\ $$$$\:\:\:\frac{\mathrm{1}}{\mathrm{4n}}\:\leqslant\:\mathrm{I}\:−\:\frac{\pi}{\mathrm{2n}^{\mathrm{3}} }\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}−\mathrm{1}} {\sum}}\:\frac{\mathrm{k}\left(\mathrm{n}−\mathrm{k}\right)}{\mathrm{sin}\left(\frac{\mathrm{k}\pi}{\mathrm{n}}\right)}\:\leqslant\:\frac{\mathrm{3}}{\mathrm{4n}} \\ $$$$\mathrm{2}.\:\mathrm{En}\:\mathrm{d}\acute {\mathrm{e}duire}\:\mathrm{la}\:\mathrm{valeur}\:\mathrm{de}\:\mathrm{I} \\ $$$$ \\…