Question Number 222151 by Ismoiljon_008 last updated on 19/Jun/25 Commented by Ismoiljon_008 last updated on 19/Jun/25 $$\:\:\:{help},\:{please} \\ $$$$ \\ $$ Commented by Rasheed.Sindhi last…
Question Number 222176 by cherokeesay last updated on 19/Jun/25 Answered by mr W last updated on 19/Jun/25 $$\angle{B}=\beta \\ $$$$\angle{A}=\mathrm{2}\beta \\ $$$$\angle{C}=\pi−\mathrm{3}\beta \\ $$$$\frac{{a}}{\mathrm{sin}\:\mathrm{2}\beta}=\frac{{c}}{\mathrm{sin}\:\mathrm{3}\beta}=\frac{\mathrm{25}}{\mathrm{sin}\:\beta} \\…
Question Number 222125 by hardmath last updated on 18/Jun/25 $$\mathrm{If}:\:\:\:\mathrm{log}_{\mathrm{3}} \left(\mathrm{5}^{\boldsymbol{\mathrm{x}}} \:+\:\frac{\mathrm{1}}{\mathrm{5}^{\boldsymbol{\mathrm{x}}} }\:+\:\mathrm{7}\right)\:\:\:\Rightarrow\:\:\:\mathrm{min}\:=\:? \\ $$ Commented by mr W last updated on 18/Jun/25 $${you}\:{mean} \\…
Question Number 222127 by Nicholas666 last updated on 18/Jun/25 $$ \\ $$$$\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\infty} \:\frac{\mathrm{ln}\left(\mathrm{tan}\left(\mathrm{tan}^{−\mathrm{1}} \left({e}^{\frac{\mathrm{1}}{\pi}\:\mathrm{tan}^{−\mathrm{1}} \:{u}} \right)\right)\right)\:}{{u}^{\mathrm{2}} \:+\:\mathrm{2}\pi{u}\:+\:\mathrm{2}\pi^{\mathrm{2}} }\:{du} \\ $$$$ \\ $$ Terms of…
Question Number 222121 by wewji12 last updated on 18/Jun/25 $$\int\:\mathrm{acos}\left(\frac{\mathrm{cos}\left(\varrho\right)}{\mathrm{1}+\mathrm{2cos}\left(\varrho\right)}\right)\:\mathrm{d}\varrho \\ $$ Answered by Nicholas666 last updated on 19/Jun/25 $$\:\:\:\:\int{cos}^{−\mathrm{1}} \:\left(\frac{{cos}\:{x}}{\mathrm{1}\:+\mathrm{2}\:{cos}\:{x}}\right)\:{dx}\:=\:\int\:\frac{{cos}\:{x}}{{cox}\left(\mathrm{1}+\mathrm{2}\:{cos}\:{x}\right)}\:{dx}\:=\int\:\frac{\mathrm{1}\:}{\mathrm{1}\:+\mathrm{2}\:{cos}\:{x}}\:{dx} \\ $$$$\:\:\:\mathrm{let};\:\:\:{t}\:=\:{tan}\:\left(\frac{{x}}{\mathrm{2}}\right)\:\Rightarrow\:{cos}\:{x}\:=\:\frac{\mathrm{1}\:−{t}^{\mathrm{2}} }{\mathrm{1}\:+\:{t}^{\mathrm{2}} }\:\mathrm{and}\:{dx}=\frac{\mathrm{2}}{\mathrm{1}\:+{t}^{\mathrm{2}}…
Question Number 222123 by mr W last updated on 18/Jun/25 Commented by mr W last updated on 18/Jun/25 $${a}\:{ball}\:{is}\:{released}\:{at}\:{the}\:{point}\:\left({d},\:{h}\right) \\ $$$${and}\:{is}\:{rebounded}\:{from}\:{the}\:{ground}\: \\ $$$${which}\:{has}\:{the}\:{shape}\:{of}\:{a}\:{hyperbola} \\ $$$${with}\:{equation}\:−\frac{{x}^{\mathrm{2}}…
Question Number 222117 by klipto last updated on 18/Jun/25 $$\boldsymbol{\mathrm{problem}}\:\mathrm{1}.\:\boldsymbol{\mathrm{for}}\:\boldsymbol{\mathrm{a}}\:\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integer}} \\ $$$$\boldsymbol{\mathrm{m}},\:\boldsymbol{\mathrm{find}}\:\boldsymbol{\mathrm{all}}\:\boldsymbol{\mathrm{triples}}\left(\boldsymbol{\mathrm{n}},\boldsymbol{\mathrm{x}},\boldsymbol{\mathrm{y}}\right)\:\boldsymbol{\mathrm{of}}\:\boldsymbol{\mathrm{positive}}\:\boldsymbol{\mathrm{integers}},\boldsymbol{\mathrm{with}} \\ $$$$\boldsymbol{\mathrm{n}}\:\boldsymbol{\mathrm{relatively}}\:\boldsymbol{\mathrm{prime}}\:\boldsymbol{\mathrm{to}}\:\boldsymbol{\mathrm{m}},\boldsymbol{\mathrm{which}}\:\boldsymbol{\mathrm{satisfy}} \\ $$$$\left(\boldsymbol{\mathrm{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}} \right)^{\boldsymbol{\mathrm{m}}} =\left(\boldsymbol{\mathrm{xy}}\right)^{\boldsymbol{\mathrm{n}}} \\ $$$$\boldsymbol{\mathrm{hint}}:\boldsymbol{\mathrm{utilize}}\:\boldsymbol{\mathrm{AM\&GM}},\boldsymbol{\mathrm{diophantine}}\:\boldsymbol{\mathrm{eqn}} \\ $$$$\boldsymbol{\mathrm{KLIPTO}}−\boldsymbol{\mathrm{QUANTA}}−\boldsymbol{\mathrm{OOZY}} \\ $$…
Question Number 222095 by wewji12 last updated on 17/Jun/25 $$\mathrm{could}\:\:\mathrm{I}\:\mathrm{consider}\:\:{Y}_{\nu} \left({z}\right)=\mathrm{cot}\left(\nu\pi\right){J}_{\nu} \left({z}\right)−\mathrm{csc}\left(\nu\pi\right){J}_{−\nu} \left({z}\right) \\ $$$$\mathrm{as}\:\infty−\infty\:\mathrm{form}\:\mathrm{limit}\:\mathrm{when}\:\nu\in\mathbb{Z} \\ $$$$\mathrm{and}\:\mathrm{How}\:\mathrm{can}\:\mathrm{i}\:\mathrm{calculate} \\ $$$${Y}_{\nu} \left({z}\right)=\mathrm{cot}\left(\nu\pi\right){J}_{\nu} \left({z}\right)−\mathrm{csc}\left(\nu\pi\right){J}_{−\nu} \left({z}\right)…?? \\ $$$$\underset{\alpha\rightarrow\nu} {\mathrm{lim}}\:\frac{\mathrm{cot}^{\mathrm{2}}…
Question Number 222104 by efronzo1 last updated on 17/Jun/25 $$\:\:\:\:\mathrm{log}\:_{\mathrm{4}} \:\mathrm{x}\:−\:\mathrm{log}\:_{\mathrm{x}^{\mathrm{2}} } \:\mathrm{8}\:=\:\mathrm{1} \\ $$$$\:\:\:\:\mathrm{x}\:=?\: \\ $$ Answered by mr W last updated on 17/Jun/25…
Question Number 222105 by alcohol last updated on 17/Jun/25 Commented by alcohol last updated on 17/Jun/25 $${please}\:{help}\:{with}\:\left({iii}\right)\:{a}\:{and}\:{b}\: \\ $$ Answered by mr W last updated…