Question Number 213250 by ajfour last updated on 01/Nov/24 Answered by a.lgnaoui last updated on 02/Nov/24 $$\mathrm{Calcul}\:\mathrm{de}\:\mathrm{l}\:\mathrm{aire} \\ $$$$ \\ $$$$\mathrm{BC}=\mathrm{a}\sqrt{\mathrm{2}}\:\:\:\:\:\:\:\mathrm{DM}=\mathrm{a}\sqrt{\mathrm{2}}\:+\frac{\mathrm{a}}{\mathrm{2}}. \\ $$$$\boldsymbol{\mathrm{S}}\left(\boldsymbol{\mathrm{ABDMC}}\right)=\mathrm{S}\left(\boldsymbol{\mathrm{ABC}}\right)+\mathrm{2}\boldsymbol{\mathrm{S}}\left(\boldsymbol{\mathrm{BDE}}\right)+\boldsymbol{\mathrm{S}}\left(\boldsymbol{\mathrm{BCEF}}\right) \\ $$$$…
Question Number 213173 by ajfour last updated on 31/Oct/24 Commented by ajfour last updated on 31/Oct/24 $${Find}\:{r} \\ $$ Answered by ajfour last updated on…
Question Number 213175 by Davidtim last updated on 31/Oct/24 $${we}\:{can}\:{find}\:{tan}\mathrm{120}\:{by}\:{tan}\left(\mathrm{180}−\mathrm{60}\right) \\ $$$${but}\:{can}\:{not}\:{find}\:{by}\:{tan}\left(\mathrm{90}+\mathrm{30}\right)\:{why}? \\ $$ Answered by efronzo1 last updated on 31/Oct/24 $$\:\mathrm{tan}\:\mathrm{120}°=\:\mathrm{tan}\:\left(\mathrm{90}°+\mathrm{30}°\right)\:=−\:\mathrm{cot}\:\mathrm{30}°=−\sqrt{\mathrm{3}} \\ $$ Answered…
Question Number 213169 by MrGaster last updated on 31/Oct/24 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove} \\ $$$$\:\:\:\:\:\underset{{n}\rightarrow\infty} {\mathrm{lim}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{n}}{\mathrm{1}+{n}^{\mathrm{2}} {x}^{\mathrm{2}} }{e}^{{x}^{\mathrm{2}} } {dx}=\frac{\pi}{\mathrm{2}}. \\ $$ Answered by Berbere last…
Question Number 213138 by Spillover last updated on 31/Oct/24 Commented by Frix last updated on 31/Oct/24 $$\mathrm{We}\:\mathrm{must}\:\mathrm{first}\:\mathrm{solve}\:\int\mathrm{sin}\:{t}^{\mathrm{3}} \:{dt}\:\mathrm{which}\:\mathrm{might} \\ $$$$\mathrm{be}\:\mathrm{possible}\:\mathrm{using} \\ $$$$\mathrm{sin}\:{t}^{\mathrm{3}} \:=\frac{\mathrm{e}^{\mathrm{i}{t}^{\mathrm{3}} } −\mathrm{e}^{−\mathrm{i}{t}^{\mathrm{3}}…
Question Number 213139 by Spillover last updated on 31/Oct/24 Answered by MrGaster last updated on 31/Oct/24 $${let}\:{u}=\sqrt{{x}}+\sqrt{{y}}+{z}\Rightarrow{dy}=\frac{\mathrm{1}}{\mathrm{2}\sqrt{{x}}}{dx}+\frac{\mathrm{1}}{\mathrm{2}\sqrt{{y}}}{dy}+\frac{\mathrm{1}}{\mathrm{2}\sqrt{{z}}}{dx} \\ $$$$\Rightarrow\mathrm{0}\leq{u}\leq\frac{\mathrm{3}\sqrt{\pi}}{\mathrm{4}} \\ $$$$\Rightarrow{dxdydz}=\mathrm{8}{u}^{\mathrm{2}} {du} \\ $$$$\Leftrightarrow\int_{\mathrm{0}} ^{\frac{\mathrm{3}\sqrt{\pi}}{\mathrm{4}}}…
Question Number 213084 by MATHEMATICSAM last updated on 30/Oct/24 $$\mathrm{If}\:\mathrm{0}\:<\:{x}\:<\:\frac{\pi}{\mathrm{2}}\:\mathrm{then}\:\mathrm{prove} \\ $$$$\mathrm{sin}\left(\mathrm{cos}{x}\right)\:<\:\mathrm{cos}{x}\:<\:\mathrm{cos}\left(\mathrm{sin}{x}\right). \\ $$$$\mathrm{not}\:\mathrm{using}\:\mathrm{graph}. \\ $$ Answered by issac last updated on 30/Oct/24 $${x}=\mathrm{acos}\left({z}\right)\:,\:{z}\in\left[\mathrm{0},\mathrm{1}\right] \\…
Question Number 213119 by a.lgnaoui last updated on 30/Oct/24 $$\mathrm{determiner}\:\boldsymbol{\mathrm{a}}\:\mathrm{et}\:\boldsymbol{\mathrm{b}}\:\mathrm{par}\:\:\:;\:\:\mathrm{AB}\:\bot\mathrm{BC} \\ $$$$\begin{cases}{\mathrm{AM}\:=\mathrm{5}}\\{\mathrm{AC}\:\:=\mathrm{16}}\end{cases} \\ $$ Commented by a.lgnaoui last updated on 30/Oct/24 Answered by A5T last…
Question Number 213081 by issac last updated on 30/Oct/24 $$\mathrm{evaluate} \\ $$$$\int_{\mathrm{0}} ^{\:\infty} \:\:\:\:\frac{\mathrm{tanh}\left(\frac{\mathrm{1}}{\mathrm{2}}{z}\right)\mathrm{csch}\left({z}\right)}{{z}}\mathrm{d}{z} \\ $$$$\left.\mathrm{1}\right)\:\mathrm{Complex}\:\mathrm{integral} \\ $$$$\left.\mathrm{2}\right)\:\mathrm{Feynman}\:\mathrm{trick} \\ $$ Answered by Berbere last updated…
Question Number 213114 by efronzo1 last updated on 30/Oct/24 $$\:\:\:\:\:\:\:\:\underline{\boldsymbol{\div}} \\ $$ Answered by issac last updated on 30/Oct/24 $$\:\:\:{f}\left({x}\right)=−{C}\left({x}−\mathrm{1}\right)+\frac{\boldsymbol{{i}}}{\pi}\left({x}−\mathrm{1}\right)^{\mathrm{5000}} \mathrm{ln}\left({x}−\mathrm{1}\right) \\ $$$$\left(\mathrm{thx}\:\mathrm{wolfram}\:\mathrm{alpha}!!\right) \\ $$$${f}\left(\mathrm{1}\right)=\mathrm{0}\:,\:\mathrm{cus}\:\:{f}\left(\mathrm{1}+\mathrm{0}\right)+{f}\left(\mathrm{1}−\mathrm{0}\right)=\mathrm{0}^{\mathrm{5000}}…