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Category: Relation and Functions

calculate-I-0-dx-1-x-2-1-x-n-and-J-0-x-n-1-x-2-1-x-n-dx-with-n-integr-gt-0-

Question Number 34263 by math khazana by abdo last updated on 03/May/18 $${calculate}\:{I}\:\:=\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\frac{{dx}}{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{{n}} \right)}\:\:{and} \\ $$$${J}\:=\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\:\:\frac{{x}^{{n}} }{\left(\mathrm{1}+{x}^{\mathrm{2}} \right)\left(\mathrm{1}+{x}^{{n}} \right)}{dx}\:{with}\:{n}\:{integr}\:>\mathrm{0} \\…

1-prove-that-x-0-1-1-1-x-lnx-x-1-2-find-2-sequences-u-n-and-v-n-u-n-k-1-n-1-ln-k-n-v-n-n-2-

Question Number 34258 by math khazana by abdo last updated on 03/May/18 $$\left.\mathrm{1}\left.\right)\:{prove}\:{that}\:\forall\:{x}\in\right]\mathrm{0},\mathrm{1}\left[\:\:\mathrm{1}−\frac{\mathrm{1}}{{x}}\leqslant{lnx}\leqslant\:{x}−\mathrm{1}\right. \\ $$$$\left.\mathrm{2}\right)\:{find}\:\mathrm{2}\:{sequences}\:{u}_{{n}} \:{and}\:{v}_{{n}} \:\:\:/ \\ $$$${u}_{{n}} \leqslant\prod_{{k}=\mathrm{1}} ^{{n}−\mathrm{1}} \:{ln}\left(\frac{{k}}{{n}}\right)\leqslant{v}_{{n}} \:\:\:\:\forall{n}\geqslant\mathrm{2} \\ $$…

f-x-f-x-3f-x-and-f-1-7-faind-f-27-

Question Number 165160 by mathlove last updated on 26/Jan/22 $${f}\left({x}+{f}\left({x}\right)\right)=\mathrm{3}{f}\left({x}\right)\:\:\:{and}\:{f}\left(−\mathrm{1}\right)=\mathrm{7} \\ $$$${faind}\:\:{f}\left(\mathrm{27}\right)=? \\ $$ Answered by Rasheed.Sindhi last updated on 26/Jan/22 $${x}=−\mathrm{1}:\:{f}\left(−\mathrm{1}+{f}\left(−\mathrm{1}\right)\right)=\mathrm{3}{f}\left(−\mathrm{1}\right) \\ $$$$\:\:\:\:\:\:{f}\left(−\mathrm{1}+\mathrm{7}\right)=\mathrm{3}\left(\mathrm{7}\right) \\…

Question-99612

Question Number 99612 by bemath last updated on 22/Jun/20 Commented by john santu last updated on 22/Jun/20 $${f}\left({x}\right)=\:\frac{\mathrm{1}}{\:\sqrt{\left[{x}\right]^{\mathrm{2}} −\left[{x}\right]−\mathrm{6}}}\:,\:\mathrm{defined}\:\mathrm{if} \\ $$$$\left[{x}\right]^{\mathrm{2}} −\left[{x}\right]−\mathrm{6}\:>\:\mathrm{0} \\ $$$$\left(\left[{x}\right]−\mathrm{3}\right)\left(\left[{x}\right]+\mathrm{2}\:\right)\:>\mathrm{0} \\…