Question Number 143057 by ERA last updated on 09/Jun/21 $$\mathrm{cos}\left(\boldsymbol{\alpha}\right)×\mathrm{cos}\left(\mathrm{2}\alpha\right)×\mathrm{cos}\left(\mathrm{4}\alpha\right)×….×\mathrm{cos}\left(\mathrm{2}^{\mathrm{n}} \boldsymbol{\alpha}\right)=\frac{\boldsymbol{\mathrm{sin}}\left(\mathrm{2}^{\boldsymbol{\mathrm{n}}+\mathrm{1}} \boldsymbol{\alpha}\right)}{\mathrm{2}^{\mathrm{n}+\mathrm{1}} \mathrm{sin}\left(\alpha\right)} \\ $$$$\boldsymbol{\mathrm{prove}} \\ $$ Answered by Dwaipayan Shikari last updated on 09/Jun/21…
Question Number 11987 by tawa last updated on 08/Apr/17 $$\mathrm{The}\:\mathrm{radius}\:\mathrm{of}\:\mathrm{the}\:\mathrm{moon}\:\mathrm{is}\:\frac{\mathrm{1}}{\mathrm{4}},\:\mathrm{and}\:\mathrm{its}\:\mathrm{mass}\:\mathrm{is}\:\:\frac{\mathrm{1}}{\mathrm{81}}\:\:\mathrm{that}\:\mathrm{of}\:\mathrm{the}\:\mathrm{earth}.\:\mathrm{If}\:\mathrm{the} \\ $$$$\mathrm{acceleration}\:\mathrm{due}\:\mathrm{to}\:\mathrm{gravity}\:\mathrm{on}\:\mathrm{the}\:\mathrm{surface}\:\mathrm{of}\:\mathrm{the}\:\mathrm{earth}\:\mathrm{is}\:\mathrm{9}.\mathrm{8m}/\mathrm{s}^{\mathrm{2}} .\:\mathrm{What} \\ $$$$\mathrm{is}\:\mathrm{its}\:\mathrm{value}\:\mathrm{on}\:\mathrm{the}\:\mathrm{moon}'\mathrm{s}\:\mathrm{surface}. \\ $$ Answered by ajfour last updated on 09/Apr/17 $${g}_{{e}}…
Question Number 77523 by MJS last updated on 07/Jan/20 $$\mathrm{how}\:\mathrm{many}\:\mathrm{silly}\:\mathrm{questions}\:\mathrm{can}\:\mathrm{a}\:\mathrm{person}\:\mathrm{ask} \\ $$$$\mathrm{within}\:\mathrm{the}\:\mathrm{first}\:\delta\:\mathrm{days}\:\mathrm{of}\:\mathrm{the}\:\mathrm{year}\:\psi\:\mathrm{when} \\ $$$$\mathrm{the}\:\mathrm{number}\:\chi_{\mathrm{0}} \:\mathrm{of}\:\mathrm{his}\:\mathrm{IDs}\notin\mathbb{R}\:\mathrm{in}\:\mathrm{the}\:\mathrm{year}\:\psi−\mathrm{1} \\ $$$$\mathrm{is}\:\mathrm{given}\:\mathrm{by}\:\left[\mathrm{ln}\:\left(\frac{\psi}{\mathrm{2}}+\mathrm{3}\mathbb{M}\right)\right]\leqslant\chi_{\mathrm{0}} <\underset{{q}_{\mathrm{0}} ,\:{x}\rightarrow\infty} {\mathrm{lim}}\frac{{q}_{\mathrm{0}} ^{{x}} \sqrt{\mathrm{2}\pi{x}}}{\mathrm{e}^{{q}_{\mathrm{0}} } } \\…
Question Number 11982 by tawa last updated on 08/Apr/17 $$\int\mathrm{x}^{\mathrm{5}} \left(\sqrt{\mathrm{x}^{\mathrm{3}} \:+\:\mathrm{1}}\right)\:\mathrm{dx} \\ $$ Answered by ajfour last updated on 08/Apr/17 $${I}=\frac{\mathrm{1}}{\mathrm{3}}\int{x}^{\mathrm{3}} \sqrt{{x}^{\mathrm{3}} +\mathrm{1}}\:\left(\mathrm{3}{x}^{\mathrm{2}} {dx}\right)…
Question Number 77514 by liki last updated on 07/Jan/20 Answered by jagoll last updated on 07/Jan/20 $${L}_{\mathrm{1}} :\:{y}=−\frac{\mathrm{3}}{\mathrm{4}}\left({x}−\mathrm{2}\right)+\left(−\mathrm{1}\right) \\ $$$${L}_{\mathrm{1}} :\:{y}\:=\:−\frac{\mathrm{3}}{\mathrm{4}}{x}+\frac{\mathrm{1}}{\mathrm{2}}\:\Rightarrow\:{a}\:=\:\frac{\mathrm{1}}{\mathrm{2}} \\ $$$${let}\:{poin}\:{C}\left({x},{y}\right)\:,\:{CA}\:{perpendicular} \\ $$$${to}\:{L}_{\mathrm{1}}…
Question Number 143048 by 0731619 last updated on 09/Jun/21 Answered by mindispower last updated on 09/Jun/21 $$=\underset{{n}\rightarrow\mathrm{2}} {\mathrm{lim}}\:\sqrt{\Gamma\left({n}\right)}−\mathrm{1}/\left(\Gamma\left({n}\right)−\mathrm{1}\right)=\frac{\mathrm{1}}{\:\sqrt{\Gamma\left({n}\right)}+\mathrm{1}} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}} \\ $$ Terms of Service…
Question Number 143051 by mnjuly1970 last updated on 09/Jun/21 $$\:\:\:\:\:\:\:_{\ast\ast\ast\ast\ast} ::\:\:{Lobachevsky}\:{Integral}\:::_{\ast\ast\ast\ast\ast} \\ $$$$\:\:\:\:\:\:\:\:\:\boldsymbol{\phi}:=\int_{\mathrm{0}} ^{\:\infty} \frac{\mathrm{s}{in}^{\mathrm{2}} \left(\:{tan}\left({x}\right)\right)}{{x}^{\:\mathrm{2}} }{dx}\overset{?} {=}\frac{\pi}{\mathrm{2}} \\ $$$$\:\:\:\:………. \\ $$ Answered by Olaf_Thorendsen…
Question Number 77513 by BK last updated on 07/Jan/20 Commented by Kunal12588 last updated on 07/Jan/20 -99631646050232081463391082912554874214147786127212261300098296284926069581800964669753771227589064527729684298473399551387739150433334227370085293404740389579109830756667470298828368996394719909762120 Commented by MJS last updated on 07/Jan/20 $$\mathrm{let}'\mathrm{s}\:\mathrm{just}\:\mathrm{give}\:\mathrm{it}\:\mathrm{a}\:\mathrm{name}…
Question Number 143045 by 0731619 last updated on 09/Jun/21 Answered by Dwaipayan Shikari last updated on 09/Jun/21 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}−{x}^{{n}} }{\mathrm{1}−{x}}{dx}=−\gamma+\psi\left({n}+\mathrm{1}\right) \\ $$$$\underset{{n}\rightarrow\mathrm{3}} {\mathrm{lim}}\frac{−\gamma−\mathrm{1}.\mathrm{833}+\psi\left({n}+\mathrm{1}\right)}{\Gamma\left({n}+\mathrm{1}\right)−\mathrm{6}}\overset{{Lhopital}} {=}\frac{\psi'\left({n}+\mathrm{1}\right)}{\Gamma'\left({n}+\mathrm{1}\right)}=\frac{\psi'\left(\mathrm{4}\right)}{\psi\left(\mathrm{4}\right)\Gamma\left(\mathrm{4}\right)}…
Question Number 143047 by 0731619 last updated on 09/Jun/21 Answered by JDamian last updated on 09/Jun/21 $$\Sigma\frac{\mathrm{1}}{{k}^{\mathrm{3}} }\:\:\mathrm{is}\:\mathrm{unknown} \\ $$ Commented by ArielVyny last updated…