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Question-196467

Question Number 196467 by sonukgindia last updated on 25/Aug/23 Answered by mr W last updated on 25/Aug/23 $${y}'\mathrm{ln}\:{y}={y} \\ $$$${y}'=\frac{{dy}}{{dx}}=\frac{{y}}{\mathrm{ln}\:{y}} \\ $$$$\mathrm{ln}\:{y}\frac{{dy}}{{y}}={dx} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\left(\mathrm{ln}\:{y}\right)^{\mathrm{2}} ={x}+{C}_{\mathrm{1}}…

prove-v-w-v-w-

Question Number 196440 by mokys last updated on 24/Aug/23 $${prove}\:\mid{v}−{w}\mid\geqslant\mid{v}\mid−\mid{w}\mid \\ $$ Answered by Frix last updated on 24/Aug/23 $${v},\:{w}\:\in\mathbb{C} \\ $$$${v}={a}\mathrm{e}^{\mathrm{i}\alpha} \wedge{w}={b}\mathrm{e}^{\mathrm{i}\beta} \wedge{a},\:{b}\:\geqslant\mathrm{0} \\…

n-m-1-1-n-m-nm-n-m-2-

Question Number 196406 by qaz last updated on 24/Aug/23 $$\underset{{n},{m}=\mathrm{1}} {\overset{\infty} {\sum}}\frac{\left(−\mathrm{1}\right)^{{n}+{m}} {nm}}{\left({n}+{m}\right)^{\mathrm{2}} }=? \\ $$ Answered by witcher3 last updated on 26/Aug/23 $$=−\underset{\mathrm{n},\mathrm{m}} {\sum}\int_{\mathrm{0}}…

Question-196429

Question Number 196429 by SANOGO last updated on 24/Aug/23 Answered by Frix last updated on 24/Aug/23 $$\mathrm{ln}\:\left(\underset{{k}=\mathrm{1}} {\overset{\infty} {\prod}}\:\left(\sqrt[{{n}}]{\mathrm{e}}\left(\frac{\mathrm{2}}{\pi}\right)^{\frac{\mathrm{1}}{{k}^{\mathrm{2}} }} \right)\right)\:=\underset{{k}=\mathrm{1}} {\overset{\infty} {\sum}}\:\left(\frac{\mathrm{ln}\:\mathrm{2}\:−\mathrm{ln}\:\pi}{{k}^{\mathrm{2}} }+\frac{\mathrm{1}}{{n}}\right)\:= \\…

Question-196388

Question Number 196388 by sonukgindia last updated on 24/Aug/23 Answered by AST last updated on 24/Aug/23 $$\phi\left(\mathrm{20}\right)=\phi\left(\mathrm{4}\right)\phi\left(\mathrm{5}\right)=\left(\mathrm{2}^{\mathrm{2}} −\mathrm{2}\right)\left(\mathrm{5}−\mathrm{1}\right)=\mathrm{8} \\ $$ Answered by BaliramKumar last updated…