Question Number 207611 by mr W last updated on 20/May/24 Answered by A5T last updated on 20/May/24 Commented by A5T last updated on 20/May/24 $$\mathrm{4}{x}+\mathrm{2}{y}={x}+\mathrm{5}{y}\Rightarrow{x}={y}\Rightarrow{s}=\mathrm{6}{x}…
Question Number 207582 by sniper237 last updated on 19/May/24 $$\int_{\mathrm{0}} ^{\pi} \:{ln}\left({sinx}\right){dx}=−\pi{ln}\mathrm{2} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\Gamma\left({x}\right){dx}\:=\:{ln}\left(\mathrm{2}\pi\right) \\ $$ Answered by mathzup last updated on 21/May/24…
Question Number 207576 by efronzo1 last updated on 19/May/24 Commented by A5T last updated on 19/May/24 $$\mathrm{4} \\ $$ Commented by mr W last updated…
Question Number 207597 by hardmath last updated on 19/May/24 Answered by A5T last updated on 20/May/24 Commented by A5T last updated on 20/May/24 $$\frac{{a}}{{x}}=\frac{{b}}{{y}}\Rightarrow\frac{{a}}{{b}}=\frac{{x}}{{y}}\Rightarrow\frac{{x}}{{x}+{y}}=\frac{{a}}{{a}+{b}} \\…
Question Number 207594 by MATHEMATICSAM last updated on 19/May/24 $$\frac{{x}\mathrm{cos}\theta}{{a}}\:+\:\frac{{y}\mathrm{sin}\theta}{{b}}\:=\:\mathrm{1} \\ $$$${x}\mathrm{sin}\theta\:−\:{y}\mathrm{cos}\theta\:=\:\sqrt{{a}^{\mathrm{2}} \mathrm{sin}^{\mathrm{2}} \theta\:+\:{b}^{\mathrm{2}} \mathrm{cos}^{\mathrm{2}} \theta} \\ $$$$\mathrm{Eliminate}\:\theta. \\ $$ Answered by Frix last updated…
Question Number 207588 by hardmath last updated on 19/May/24 $$\sqrt{\mathrm{2}}\:\mathrm{sin}\boldsymbol{\mathrm{x}}\:\:+\:\:\mathrm{cos}\boldsymbol{\mathrm{x}}\:\:\geqslant\:\:\mathrm{1} \\ $$ Answered by mr W last updated on 19/May/24 $$\frac{\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{3}}}\:\mathrm{sin}\:{x}+\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}}\:\mathrm{cos}\:{x}\geqslant\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}} \\ $$$$\mathrm{sin}\:\alpha\:\:\mathrm{sin}\:{x}+\mathrm{cos}\:\alpha\:\mathrm{cos}\:{x}\geqslant\frac{\mathrm{1}}{\:\sqrt{\mathrm{3}}} \\ $$$${with}\:\alpha=\mathrm{cos}^{−\mathrm{1}}…
Question Number 207590 by hardmath last updated on 19/May/24 $$\mathrm{x}^{\:\boldsymbol{\mathrm{lg}}\:\boldsymbol{\mathrm{x}}} \:\:<\:\:\mathrm{10}^{\mathrm{4}} \\ $$$$\mathrm{The}\:\mathrm{number}\:\mathrm{of}\:\mathrm{roots}? \\ $$ Commented by hardmath last updated on 19/May/24 $$\mathrm{professor},\:\mathrm{integer}\:\mathrm{roots} \\ $$…
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Question Number 207546 by pete last updated on 18/May/24 $$\mathrm{The}\:\mathrm{real}\:\mathrm{roots}\:\mathrm{of}\:\mathrm{the}\:\mathrm{equation}\:\mathrm{x}^{\mathrm{2}} +\mathrm{6x}+\mathrm{c}=\mathrm{0} \\ $$$$\mathrm{differ}\:\mathrm{by}\:\mathrm{2n},\:\mathrm{where}\:\mathrm{n}\:\mathrm{is}\:\mathrm{a}\:\mathrm{real}\:\mathrm{non}−\mathrm{zero}. \\ $$$$\mathrm{Show}\:\mathrm{that}\:\mathrm{n}^{\mathrm{2}} =\mathrm{9}−\mathrm{c} \\ $$$$\mathrm{Given}\:\mathrm{that}\:\mathrm{the}\:\mathrm{roots}\:\mathrm{also}\:\mathrm{have}\:\mathrm{opposite} \\ $$$$\mathrm{signs},\:\mathrm{find}\:\mathrm{the}\:\mathrm{set}\:\mathrm{of}\:\mathrm{possible}\:\mathrm{values}\:\mathrm{of}\:\mathrm{n} \\ $$ Answered by A5T…
Question Number 207547 by SANOGO last updated on 18/May/24 $${lim}\rightarrow+{oo}\:\int_{\mathrm{1}} ^{+{oo}} \:\frac{{x}^{{n}} }{\mathrm{1}+{x}^{{n}+\mathrm{2}} }{dx}\:=? \\ $$ Commented by SWPlaysMC last updated on 18/May/24 $$\mathrm{Just}\:\mathrm{so}\:\mathrm{you}\:\mathrm{know},\:\mathrm{the}\:\mathrm{infinity}\:\mathrm{symbol}\:\mathrm{is}\:\infty\:\mathrm{and}\:\mathrm{not}\:“{oo}'' \\…