Question Number 206251 by TonyCWX08 last updated on 10/Apr/24 $$\mathrm{3}{x}^{\mathrm{2}} −\mathrm{12}{x}−\mathrm{5}\sqrt{{x}^{\mathrm{2}} −\mathrm{4}{x}−\mathrm{1}}−\mathrm{5}=\mathrm{0} \\ $$ Answered by A5T last updated on 10/Apr/24 $$\mathrm{3}\left({x}^{\mathrm{2}} −\mathrm{4}{x}−\mathrm{1}\right)−\mathrm{2}−\mathrm{5}\sqrt{{x}^{\mathrm{2}} −\mathrm{4}{x}−\mathrm{1}}=\mathrm{0} \\…
Question Number 206244 by cortano21 last updated on 10/Apr/24 $$\:\:\:\zeta \\ $$ Answered by A5T last updated on 10/Apr/24 $$\pi{r}_{{a}} ^{\mathrm{3}} =\pi{r}_{{b}} ^{\mathrm{3}} ×\frac{\mathrm{1}}{\mathrm{2}}\Rightarrow\frac{{r}_{{a}} }{{r}_{{b}}…
Question Number 206273 by EmGent last updated on 10/Apr/24 $$\mathrm{Does}\:\mathrm{anyone}\:\mathrm{know}\:\mathrm{how}\:\mathrm{this}\:\mathrm{works}\:? \\ $$$$ \\ $$$$\mathrm{I}\:\mathrm{have}\:{d}\psi\:=\:\left({x}^{\mathrm{2}} -{cy}^{\mathrm{2}} \right){dy} \\ $$$$ \\ $$$$\mathrm{And}\:\mathrm{my}\:\mathrm{physics}\:\mathrm{teacher}\:\mathrm{says}\:\mathrm{it}\:\mathrm{is}\:\left(\mathrm{or}\:\mathrm{can}\right. \\ $$$$\left.\mathrm{be}\right)\:\mathrm{a}\:\mathrm{harmonic}\:\mathrm{function}\:\left(\Delta\psi\:=\:\mathrm{0}\right) \\ $$$$\mathrm{Can}\:\mathrm{anyone}\:\mathrm{explain}\:? \\…
Question Number 206232 by mr W last updated on 09/Apr/24 Answered by A5T last updated on 09/Apr/24 $${S}_{\mathrm{2}} ={S}_{\mathrm{1}} +{S}_{\mathrm{3}} −\mathrm{2}{AB}×{BCcos}\left(\theta\right) \\ $$$${S}_{\mathrm{1}} ={S}_{\mathrm{2}} +{S}_{\mathrm{3}}…
Question Number 206216 by cortano21 last updated on 09/Apr/24 Answered by A5T last updated on 09/Apr/24 $${HC}=\sqrt{\mathrm{108}+\mathrm{4}}=\sqrt{\mathrm{112}}=\mathrm{4}\sqrt{\mathrm{7}} \\ $$$${Let}\:{HC}\:{meet}\:{the}\:{circumcircle}\:{of}\:{the}\:{hexagon} \\ $$$${at}\:{L}\:{then},\:{HC}×{HL}={FH}×{HE}=\mathrm{8}\Rightarrow{HL}=\frac{\mathrm{2}\sqrt{\mathrm{7}}}{\mathrm{7}} \\ $$$${sin}\angle{EHC}=\frac{\sqrt{\mathrm{108}}}{\:\sqrt{\mathrm{112}}}=\frac{\mathrm{3}\sqrt{\mathrm{21}}}{\mathrm{14}}\Rightarrow{cos}\angle{EHC}=\frac{\sqrt{\mathrm{7}}}{\mathrm{14}} \\ $$$$\Rightarrow\sqrt{{x}^{\mathrm{2}}…
Question Number 206200 by Shrodinger last updated on 09/Apr/24 $$\int\frac{{xsinx}}{\mathrm{1}−{cosx}}{dx} \\ $$ Answered by Frix last updated on 09/Apr/24 $$\int\frac{{x}\mathrm{sin}\:{x}}{\mathrm{1}−\mathrm{cos}\:{x}}{dx}=\mathrm{i}\int\frac{{x}\left(\mathrm{e}^{\mathrm{i}{x}} +\mathrm{1}\right)}{\mathrm{e}^{\mathrm{i}{x}} −\mathrm{1}}{dx}= \\ $$$$=\mathrm{i}\int{xdx}+\mathrm{2i}\int\frac{{x}}{\mathrm{e}^{\mathrm{i}{x}} −\mathrm{1}}{dx}…
Question Number 206212 by cortano21 last updated on 09/Apr/24 Commented by Frix last updated on 09/Apr/24 $${f}'\left(\mathrm{0}\right)={c}\:\mathrm{which}\:\mathrm{is}\:\mathrm{independent}\:\mathrm{of}\:{a} \\ $$ Commented by TheHoneyCat last updated on…
Question Number 206230 by BaliramKumar last updated on 09/Apr/24 $$\mathrm{Expand}\:\:\:\:{x}^{\mathrm{2}} \:+\:\mathrm{2}{x}\:+\:\mathrm{3}\:\:\:{respect}\:{to}\:{x}\:=\:−\mathrm{2}. \\ $$$$\left(\mathrm{a}\right)\:\left({x}\:−\:\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2}\left({x}\:+\:\mathrm{2}\right)\:+\:\mathrm{3} \\ $$$$\left(\mathrm{b}\right)\:\left({x}\:+\:\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2}\left({x}\:+\:\mathrm{2}\right)\:+\:\mathrm{3} \\ $$$$\left(\mathrm{c}\right)\:\left({x}\:+\:\mathrm{2}\right)^{\mathrm{2}} +\:\mathrm{2}\left({x}\:+\:\mathrm{2}\right)\:+\:\mathrm{3} \\ $$$$\left(\mathrm{d}\right)\:\left({x}\:−\:\mathrm{2}\right)^{\mathrm{2}} −\mathrm{2}\left({x}\:−\:\mathrm{2}\right)\:−\:\mathrm{3} \\ $$$$…
Question Number 206198 by lmcp1203 last updated on 09/Apr/24 Answered by A5T last updated on 09/Apr/24 $$\frac{{sin}\mathrm{4}\theta}{{AD}}=\frac{{sin}\left(\mathrm{90}−\mathrm{3}\theta\right)}{{AB}};\frac{{sin}\left(\theta\right)}{{AD}}=\frac{{sin}\left(\mathrm{90}−\mathrm{4}\theta\right)}{{CD}={AB}} \\ $$$$\Rightarrow\frac{{sin}\left(\mathrm{4}\theta\right)}{{sin}\left(\mathrm{90}−\mathrm{3}\theta\right)}=\frac{{sin}\left(\theta\right)}{{sin}\left(\mathrm{90}−\mathrm{4}\theta\right)}\Rightarrow{x}=\mathrm{20}° \\ $$ Answered by lmcp1203 last…
Question Number 206224 by mathzup last updated on 09/Apr/24 $${find}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:{arctan}\left({x}^{\mathrm{5}} \right){dx} \\ $$ Answered by Frix last updated on 10/Apr/24 $$\int\mathrm{tan}^{−\mathrm{1}} \:{x}^{{n}} \:{dx}\:\underset{{u}'=\mathrm{1}\wedge{v}=\mathrm{tan}^{−\mathrm{1}}…