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Category: Coordinate Geometry

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

lim-x-0-1-cosxcos2x-cos-nx-x-2-n-n-1-2n-1-12-

Question Number 197359 by sniper237 last updated on 14/Sep/23 $$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\:\frac{\mathrm{1}−{cosxcos}\mathrm{2}{x}…{cos}\left({nx}\right)}{{x}^{\mathrm{2}} }\:=\:\frac{{n}\left({n}+\mathrm{1}\right)\left(\mathrm{2}{n}+\mathrm{1}\right)}{\mathrm{12}}\: \\ $$ Commented by universe last updated on 16/Sep/23 Answered by witcher3 last…

If-A-a-2-2a-B-1-a-2-2-a-and-S-1-0-are-three-points-then-prove-that-1-SA-1-SB-1-

Question Number 196992 by MATHEMATICSAM last updated on 05/Sep/23 $$\mathrm{If}\:\mathrm{A}\left({a}^{\mathrm{2}} ,\:\mathrm{2}{a}\right),\:\mathrm{B}\left(\frac{\mathrm{1}}{{a}^{\mathrm{2}} },\:\frac{−\:\mathrm{2}}{{a}}\right)\:\mathrm{and}\:\mathrm{S}\left(\mathrm{1},\:\mathrm{0}\right)\: \\ $$$$\mathrm{are}\:\mathrm{three}\:\mathrm{points}\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}, \\ $$$$\frac{\mathrm{1}}{\mathrm{SA}}\:+\:\frac{\mathrm{1}}{\mathrm{SB}}\:=\:\mathrm{1}. \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com