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Author: Tinku Tara

Question-201817

Question Number 201817 by Calculusboy last updated on 13/Dec/23 Answered by witcher3 last updated on 13/Dec/23 $$=\int\left(\mathrm{sin}\left(\mathrm{ln}\left(\mathrm{x}+\mathrm{1}\right)\right)\left(\sqrt{\mathrm{x}+\mathrm{1}}−\mathrm{1}\right)\right)^{\mathrm{2}} \mathrm{dx} \\ $$$$=\int\mathrm{sin}^{\mathrm{2}} \left(\mathrm{ln}\left(\mathrm{x}+\mathrm{1}\right)\right)\left(\mathrm{x}+\mathrm{2}−\mathrm{2}\sqrt{\mathrm{x}+\mathrm{1}}\right) \\ $$$$\mathrm{ln}\left(\sqrt{\mathrm{x}+\mathrm{1}}\right)=\mathrm{t} \\ $$$$\mathrm{x}+\mathrm{1}=\mathrm{e}^{\mathrm{2t}}…

If-xyz-R-xyz-1-prove-that-the-following-inequality-holds-x-2x-5-x-4-y-2y-5-y-4-z-2z-5-z-4-3-7-Solution-please-with-an-advice-to-get-better-at-inequalities-and-which-book-wou

Question Number 201819 by York12 last updated on 13/Dec/23 $$\mathrm{If}\:{xyz}\:\in\mathbb{R}^{+} \:,\:{xyz}=\mathrm{1}\:,\:\mathrm{prove}\:\mathrm{that}\:\mathrm{the}\:\mathrm{following}\:\mathrm{inequality}\:\mathrm{holds}: \\ $$$$\frac{{x}}{\mathrm{2}{x}^{\mathrm{5}} +{x}+\mathrm{4}}+\frac{{y}}{\mathrm{2}{y}^{\mathrm{5}} +{y}+\mathrm{4}}+\frac{{z}}{\mathrm{2}{z}^{\mathrm{5}} +{z}+\mathrm{4}}\geqslant\frac{\mathrm{3}}{\mathrm{7}}. \\ $$$$\boldsymbol{\mathrm{Solution}}\:\boldsymbol{\mathrm{please}}\:\boldsymbol{\mathrm{with}}\:\boldsymbol{\mathrm{an}}\:\boldsymbol{\mathrm{advice}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{get}}\:\boldsymbol{\mathrm{better}} \\ $$$$\boldsymbol{\mathrm{at}}\:\boldsymbol{\mathrm{inequalities}}\:\boldsymbol{\mathrm{and}}\:\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{book}}\:\boldsymbol{\mathrm{would}}\:\boldsymbol{\mathrm{you}}\:\boldsymbol{\mathrm{recommend}}. \\ $$$$\boldsymbol{\mathrm{Thanks}}\:\boldsymbol{\mathrm{in}}\:\boldsymbol{\mathrm{advance}}! \\ $$$$\: \\…

shortest-distance-from-6-0-to-x-2-y-2-16-0-

Question Number 201829 by 281981 last updated on 13/Dec/23 $${shortest}\:{distance}\:{from}\:\left(−\mathrm{6},\mathrm{0}\right){to}\:{x}^{\mathrm{2}} −{y}^{\mathrm{2}} +\mathrm{16}=\mathrm{0} \\ $$ Answered by esmaeil last updated on 13/Dec/23 $${d}=\sqrt{\left({x}+\mathrm{6}\right)^{\mathrm{2}} +{y}^{\mathrm{2}} }=\sqrt{\overset{\mathrm{2}} {{x}}+\mathrm{12}{x}+\mathrm{36}+\overset{\mathrm{2}}…