Question Number 154052 by mathdanisur last updated on 13/Sep/21 $$\mathrm{Prove}\:\mathrm{without}\:\mathrm{any}\:\mathrm{software} \\ $$$$\underset{\:\mathrm{2}−\sqrt{\mathrm{3}}} {\overset{\:\mathrm{1}} {\int}}\:\mathrm{e}^{−\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\mathrm{dx}\:<\:\frac{\boldsymbol{\pi}}{\mathrm{6}}\:\:\mathrm{and}\underset{\:\mathrm{1}} {\overset{\:\mathrm{2}+\sqrt{\mathrm{3}}} {\int}}\mathrm{e}^{−\boldsymbol{\mathrm{x}}^{\mathrm{2}} } \:\mathrm{dx}\:<\:\frac{\boldsymbol{\pi}}{\mathrm{6}} \\ $$ Terms of Service…
Question Number 154045 by EDWIN88 last updated on 14/Sep/21 $$\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\:\sqrt[{{n}}]{\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{{k}^{\mathrm{2}} }{\mathrm{2}{k}^{\mathrm{2}} −\mathrm{2}{nk}+{n}^{\mathrm{2}} }\right)\left(\underset{{k}=\mathrm{1}} {\overset{{n}} {\sum}}\:\frac{{k}^{\mathrm{2}} }{\mathrm{3}{k}^{\mathrm{2}} −\mathrm{3}{nk}+{n}^{\mathrm{2}} }\right)}\:=? \\ $$ Commented by…
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Question Number 154036 by liberty last updated on 13/Sep/21 $$\:\mathrm{49}\left(\frac{{x}+\mathrm{5}}{{x}−\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{36}\left(\frac{{x}+\mathrm{5}}{{x}−\mathrm{1}}\right)^{\mathrm{2}} =\:\mathrm{85} \\ $$ Answered by Rasheed.Sindhi last updated on 13/Sep/21 $$ \\ $$$$\:\mathrm{49}\left(\frac{{x}+\mathrm{5}}{{x}−\mathrm{2}}\right)^{\mathrm{2}} +\mathrm{36}\left(\frac{{x}+\mathrm{5}}{{x}−\mathrm{1}}\right)^{\mathrm{2}}…
Question Number 154034 by amin96 last updated on 13/Sep/21 $${prove}\:{that}\: \\ $$$${n}+{x}=\sqrt{{n}^{\mathrm{2}} +{x}\sqrt{{n}^{\mathrm{2}} +\left({x}+{n}\right)\sqrt{{n}^{\mathrm{2}} +\left({x}+\mathrm{2}{n}\right)\sqrt{{n}^{\mathrm{2}} \ldots}}}} \\ $$$${Ramanujan}'{s}\:{nested}\:{radikal} \\ $$ Answered by mr W last…
Question Number 88491 by M±th+et£s last updated on 11/Apr/20 $$\boldsymbol{{solve}} \\ $$$${cos}\left({x}\right)={k} \\ $$ Answered by mr W last updated on 11/Apr/20 $${if}\:\mid{k}\mid\leqslant\mathrm{1}: \\ $$$$\Rightarrow{x}=\mathrm{2}{n}\pi\pm\mathrm{cos}^{−\mathrm{1}}…
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Question Number 154012 by mathdanisur last updated on 13/Sep/21 $$\mathrm{Determine}\:\mathrm{all}\:\mathrm{the}\:\mathrm{perfect}\:\mathrm{squares}\:\mathrm{on} \\ $$$$\mathrm{form}\:\:\boldsymbol{\mathrm{p}}^{\boldsymbol{\mathrm{n}}} \:+\:\mathrm{1}\:\:\mathrm{where}\:\:\boldsymbol{\mathrm{p}}\:\:\mathrm{is}\:\mathrm{a}\:\mathrm{prime}\:\mathrm{number} \\ $$$$\mathrm{and}\:\:\boldsymbol{\mathrm{n}}\:\:\mathrm{a}\:\mathrm{positive}\:\mathrm{integer}. \\ $$ Commented by Rasheed.Sindhi last updated on 13/Sep/21 $$\mathrm{For}\:\mathrm{some}\:\mathrm{cases}…
Question Number 154013 by mathdanisur last updated on 13/Sep/21 $$\mathrm{Prove}\:\mathrm{without}\:\mathrm{any}\:\mathrm{software} \\ $$$$\underset{\:\frac{\mathrm{1}}{\mathrm{2}}} {\overset{\:\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}} {\int}}\:\frac{\mathrm{1}}{\mathrm{x}}\:\mathrm{log}\left(\mathrm{1}+\mathrm{2x}^{\mathrm{2}} +\mathrm{x}^{\mathrm{4}} \right)\mathrm{dx}\:<\:\sqrt{\mathrm{7}}\:-\:\sqrt{\mathrm{5}} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 154015 by mathdanisur last updated on 13/Sep/21 $$\mathrm{If}\:\:\mathrm{x};\mathrm{y};\mathrm{z}>\mathrm{0}\:\:\mathrm{then}\:\mathrm{prove}\:\mathrm{that}: \\ $$$$\frac{\mathrm{x}}{\mathrm{x}^{\mathrm{2}} +\mathrm{yz}}\:+\:\frac{\mathrm{y}}{\mathrm{y}^{\mathrm{2}} +\mathrm{zx}}\:+\:\frac{\mathrm{z}}{\mathrm{z}^{\mathrm{2}} +\mathrm{xy}}\:\leqslant\:\frac{\mathrm{x}^{\mathrm{2}} +\mathrm{y}^{\mathrm{2}} +\mathrm{z}^{\mathrm{2}} }{\mathrm{2xyz}} \\ $$ Terms of Service Privacy Policy…