Question Number 209347 by RoseAli last updated on 07/Jul/24 Answered by Berbere last updated on 07/Jul/24 $$\int_{−\mathrm{4}} ^{\mathrm{4}} {f}\left({x}\right){dx}=\mathrm{2}\underset{\mathrm{0}} {\int}^{\mathrm{4}} {f}\left({x}^{\mathrm{2}} \right){dx} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{4}}…
Question Number 209308 by Erico last updated on 06/Jul/24 $$\mathrm{Donner}\:\mathrm{l}'\acute {\mathrm{e}quivalence}\:\mathrm{simple} \\ $$$$\mathrm{de}\:\mathrm{I}_{\mathrm{n}} =\underset{\:\mathrm{0}} {\int}^{\:\mathrm{1}} \frac{{t}^{{n}} }{{t}^{{n}} −{t}+\mathrm{1}}{dt} \\ $$ Answered by mathzup last updated…
Question Number 209309 by hardmath last updated on 06/Jul/24 $$\mathrm{m}\:,\:\mathrm{n}\:\in\:\mathbb{N} \\ $$$$\mathrm{m}\:\geqslant\:\mathrm{2}\:\:\:\mathrm{and}\:\:\:\mathrm{n}\:\geqslant\:\mathrm{2} \\ $$$$\mathrm{p}\:>\:\mathrm{0}\:\:\:\mathrm{and}\:\:\:\mathrm{q}\:>\:\mathrm{0} \\ $$$$\mathrm{p}\:+\:\mathrm{q}\:=\:\mathrm{1} \\ $$$$\mathrm{Prove}\:\mathrm{that}:\:\:\:\left(\mathrm{1}−\mathrm{q}^{\boldsymbol{\mathrm{n}}} \right)^{\boldsymbol{\mathrm{m}}} \:+\:\left(\mathrm{1}−\mathrm{p}^{\boldsymbol{\mathrm{m}}} \right)^{\boldsymbol{\mathrm{n}}} \:\geqslant\:\mathrm{1} \\ $$ Terms…
Question Number 209304 by RoseAli last updated on 06/Jul/24 Answered by A5T last updated on 06/Jul/24 $${a}^{{x}} +{b}^{{x}} =\mathrm{62} \\ $$$${a}+{b}=\mathrm{8};{ab}=\mathrm{1} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\left({a}+{b}\right)^{\mathrm{2}}…
Question Number 209307 by efronzo1 last updated on 06/Jul/24 Commented by mr W last updated on 06/Jul/24 $$\mid{QR}\mid<\frac{\mathrm{8}}{\mathrm{3}}−\mathrm{1}=\frac{\mathrm{5}}{\mathrm{3}} \\ $$$$\frac{{QR}}{{PR}}=\frac{\mathrm{2}}{\mathrm{1}}\:\Rightarrow\frac{{PQ}}{{QR}}=\frac{\sqrt{\mathrm{5}}}{\mathrm{2}} \\ $$$$\mid{PQ}\mid=\frac{\sqrt{\mathrm{5}}}{\mathrm{2}}×{QR}<\frac{\sqrt{\mathrm{5}}}{\mathrm{2}}×\frac{\mathrm{5}}{\mathrm{3}}=\frac{\mathrm{5}\sqrt{\mathrm{5}}}{\mathrm{6}}<\sqrt{\mathrm{5}} \\ $$$$\Rightarrow{impossible}\:{that}\:\mid{PQ}\mid=\sqrt{\mathrm{5}}\:! \\…
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Question Number 209288 by Jubr last updated on 06/Jul/24 Answered by A5T last updated on 06/Jul/24 $$\frac{{x}}{{r}}=\frac{\mathrm{8}}{{r}}\Rightarrow{x}=\mathrm{8} \\ $$$${r}^{\mathrm{2}} =\left(\mathrm{2}{x}\right)^{\mathrm{2}} +\left({r}−\mathrm{8}\right)^{\mathrm{2}} \Rightarrow{r}=\mathrm{20} \\ $$$$\left[{Yellow}\right]=\frac{\mathrm{90}}{\mathrm{360}}×\mathrm{400}\pi−\frac{\mathrm{20}×\mathrm{20}}{\mathrm{2}}=\mathrm{100}\pi−\mathrm{200} \\…
Question Number 209289 by Jubr last updated on 06/Jul/24 Answered by A5T last updated on 06/Jul/24 $$\frac{{s}}{{x}}=\frac{\mathrm{7}}{\mathrm{21}}\Rightarrow{x}=\mathrm{3}{s} \\ $$$$\frac{{s}}{{y}}=\frac{\mathrm{14}}{\mathrm{21}}\Rightarrow{y}=\frac{\mathrm{3}{s}}{\mathrm{2}} \\ $$$${x}^{\mathrm{2}} +{y}^{\mathrm{2}} =\mathrm{21}^{\mathrm{2}} \Rightarrow\mathrm{9}{s}^{\mathrm{2}} +\frac{\mathrm{9}{s}^{\mathrm{2}}…
Question Number 209290 by klipto last updated on 06/Jul/24 $$\boldsymbol{\mathrm{a}}^{\mathrm{2}} −\boldsymbol{\mathrm{a}}−^{\mathrm{1000}} \sqrt{\left(\mathrm{1}+\mathrm{8000}\boldsymbol{\mathrm{a}}\right)}=\mathrm{1000} \\ $$$$\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{a}} \\ $$ Commented by mr W last updated on 06/Jul/24 $${you}\:{can}\:{only}\:{approximate}!…