Question Number 12547 by @ANTARES_VY last updated on 25/Apr/17 $$\boldsymbol{\mathrm{This}} \\ $$$$\boldsymbol{\mathrm{y}}=\boldsymbol{\mathrm{sin}}\frac{\boldsymbol{\mathrm{x}}}{\mathrm{2}}\:\:\boldsymbol{\mathrm{find}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{range}}\:\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{the}}\:\: \\ $$$$\boldsymbol{\mathrm{function}}. \\ $$ Answered by mrW1 last updated on 25/Apr/17 $${x},{y}\in\mathbb{R} \\…
Question Number 12543 by @ANTARES_VY last updated on 25/Apr/17 $$\boldsymbol{\mathrm{This}} \\ $$$$\boldsymbol{\mathrm{x}}^{\mathrm{2}} −\boldsymbol{\alpha\mathrm{x}}+\boldsymbol{\alpha}−\mathrm{1}=\mathrm{0}. \\ $$$$\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{roots}}\:\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{equation}}\:\:\boldsymbol{\mathrm{x}}_{\mathrm{1}} \:\:\boldsymbol{\mathrm{and}}\:\:\boldsymbol{\mathrm{x}}_{\mathrm{2}} \\ $$$$\boldsymbol{\mathrm{a}}\:\:\boldsymbol{\mathrm{what}}'\boldsymbol{\mathrm{s}}\:\:\boldsymbol{\mathrm{the}}\:\:\boldsymbol{\mathrm{value}}\:\:\boldsymbol{\mathrm{of}}\:\:\boldsymbol{\mathrm{x}}_{\mathrm{1}} ^{\mathrm{2}} +\boldsymbol{\mathrm{x}}_{\mathrm{2}} ^{\mathrm{2}} \:\:\boldsymbol{\mathrm{this}}\:\:\boldsymbol{\mathrm{collection}}\:\:\boldsymbol{\mathrm{of}} \\ $$$$\boldsymbol{\mathrm{smille}}\left(\boldsymbol{\mathrm{minimum}}\right)\:\:\boldsymbol{\mathrm{value}}. \\…
Question Number 143609 by mnjuly1970 last updated on 16/Jun/21 $$\:\:\: \\ $$$$\:{Prove}\:{that}:: \\ $$$$\:\:\Omega:=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{{ln}^{\mathrm{2}} \left(\mathrm{1}−{x}\right).{ln}\left({x}\right)}{{x}}{dx}=\frac{−\mathrm{1}}{\mathrm{2}}\:\zeta\:\left(\mathrm{4}\:\right) \\ $$$${Without}\:{using}\:{the}\:“{Beta}\:{function}'' \\ $$$$\:\:{m}.{n} \\ $$ Answered by…
Question Number 78046 by jagoll last updated on 13/Jan/20 $${minimum}\:{of}\: \\ $$$${function}\:{y}\:=\:\sqrt{{x}^{\mathrm{2}} +{e}^{\mathrm{2}{x}} }\:\:{is} \\ $$ Answered by john santu last updated on 13/Jan/20 $${y}'\:=\:\frac{\mathrm{2}{x}+\mathrm{2}{e}^{\mathrm{2}{x}}…
Question Number 143505 by mim24 last updated on 15/Jun/21 Commented by mr W last updated on 15/Jun/21 $${f}\left({x}\right)=\frac{{x}}{\left({x}+\mathrm{2}+\sqrt{\mathrm{3}}\right)\left({x}+\mathrm{2}−\sqrt{\mathrm{3}}\right)} \\ $$$${f}\left({x}\right)\rightarrow\pm\infty\:{when}\:{x}\rightarrow−\mathrm{2}\pm\sqrt{\mathrm{3}} \\ $$$$\Rightarrow{no}\:{maximum},\:{no}\:{minimum} \\ $$ Answered…
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Question Number 143461 by Willson last updated on 14/Jun/21 $$\mathrm{Prove}\:\mathrm{that}\::\:\forall\mathrm{n}\in\mathbb{N}^{\ast} \\ $$$$\mathrm{a}.\:\:\:\:\:\:\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}{C}_{\mathrm{n}} ^{\mathrm{k}} \frac{\left(−\mathrm{1}\right)^{\mathrm{k}} }{\mathrm{k}}\:=\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}\frac{\mathrm{1}}{\mathrm{k}} \\ $$$$\mathrm{b}.\:\:\:\:\:\:\underset{\mathrm{k}=\mathrm{1}} {\overset{\mathrm{n}} {\sum}}{C}_{\mathrm{n}} ^{\mathrm{k}} \:\frac{\left(−\mathrm{1}\right)^{\mathrm{k}}…
Question Number 143454 by mnjuly1970 last updated on 14/Jun/21 $$ \\ $$$$\:\:\:……..{nice}\:…….{integral}……. \\ $$$$\:\:\mathscr{T}\:\::=\int_{\mathrm{0}} ^{\:\infty} \frac{{arctan}\left({x}\right)}{{x}^{\:{ln}\left({x}\right)\:+\mathrm{1}} }{dx}\overset{?} {=}\frac{\pi\sqrt{\pi}}{\mathrm{4}} \\ $$$$ \\ $$ Answered by mindispower…
Question Number 143429 by mnjuly1970 last updated on 14/Jun/21 $$\:\: \\ $$$$\:\:\:\:\:\:\:…….\:{nice}\:…..{integral}……. \\ $$$$\:\:\:\:{Evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\xi\::=\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{{ln}\left(\mathrm{1}−{t}\right)}{\mathrm{1}+{t}^{\mathrm{2}} }\:{dt}\:=? \\ $$ Answered by Dwaipayan Shikari…
Question Number 77842 by jagoll last updated on 11/Jan/20 $${given}\: \\ $$$${f}\left({x}\right)=\:{x}^{\mathrm{2}} +\mathrm{sin}\left(\mathrm{2}{x}\right)\:+\:\int\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{4}}} {\:}}{f}\left({x}\right){dx} \\ $$$${f}\left(\frac{\pi}{\mathrm{2}}\right)=? \\ $$ Answered by mr W last updated…