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Question Number 205371 by 073 last updated on 19/Mar/24 Commented by lepuissantcedricjunior last updated on 19/Mar/24 $$\boldsymbol{\mathrm{I}}=\int_{\mathrm{0}} ^{\mathrm{2}} \boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\mathrm{8}\right)\boldsymbol{\mathrm{dx}} \\ $$$$\boldsymbol{\mathrm{posons}}\:\begin{cases}{\boldsymbol{\mathrm{u}}=\boldsymbol{\mathrm{log}}\left(\boldsymbol{\mathrm{x}}^{\mathrm{3}} +\mathrm{8}\right)}\\{\boldsymbol{\mathrm{v}}'=\mathrm{1}}\end{cases}\Leftrightarrow\begin{cases}{\boldsymbol{\mathrm{u}}'=\frac{\mathrm{3}\boldsymbol{{x}}^{\mathrm{2}} }{\left(\boldsymbol{{x}}^{\mathrm{3}} +\mathrm{8}\right)\boldsymbol{\mathrm{ln}}\mathrm{10}}}\\{\boldsymbol{\mathrm{v}}=\boldsymbol{\mathrm{x}}}\end{cases}…
Question Number 205380 by cortano12 last updated on 19/Mar/24 Commented by Rasheed.Sindhi last updated on 19/Mar/24 $$\boldsymbol{{please}}\:{don}'{t}\:{write}\:{your}\:{solution} \\ $$$${as}\:{a}\:{comment}!\:{Instead}\:{write}\:{it}\:{as} \\ $$$${answer}\:\boldsymbol{{in}}\:\boldsymbol{{order}}\:\boldsymbol{{to}}\:\boldsymbol{{prserve}}\:\boldsymbol{{chronological}} \\ $$$$\boldsymbol{{order}}\:\boldsymbol{{of}}\:\boldsymbol{{answers}}! \\ $$…
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Question Number 205394 by Red1ight last updated on 19/Mar/24 $$\mathrm{let}\:{x}\:\mathrm{and}\:{y}\:\mathrm{be}\:\mathrm{random}\:\mathrm{numbers}\:\mathrm{from}\:\mathrm{0}\:\mathrm{to}\:\mathrm{10} \\ $$$$\mathrm{where}\:{x},{y}\in\mathbb{R} \\ $$$$\mid{x}−{y}\mid\geqslant{d} \\ $$$$\mathrm{what}\:\mathrm{is}\:\mathrm{the}\:\mathrm{probability}\:\mathrm{that}\:\mathrm{their}\:\mathrm{sum}\:\mathrm{is}\:\mathrm{less}\:\mathrm{than}\:\mathrm{10} \\ $$$$\mathrm{in}\:\mathrm{the}\:\mathrm{following}\:\mathrm{cases} \\ $$$$\left.{a}\right)\:{d}=\mathrm{0} \\ $$$$\left.{b}\right)\:{d}=\mathrm{1} \\ $$$$\left.{c}\right)\:{d}=\mathrm{2} \\…
Question Number 205379 by cortano12 last updated on 19/Mar/24 Answered by MM42 last updated on 19/Mar/24 $${hop}\rightarrow={lim}_{{x}\rightarrow\mathrm{0}} \frac{−{cosx}×{sin}\left({sinx}\right)+{sinx}}{\mathrm{4}{x}^{\mathrm{3}} } \\ $$$${hop}\rightarrow=\frac{{sinx}×{sin}\left({sinx}\right)−{cos}^{\mathrm{2}} {x}×{cos}\left({sinx}\right)+{cosx}}{\mathrm{12}{x}^{\mathrm{2}} } \\ $$$$={lim}_{{x}\rightarrow\mathrm{0}}…
Question Number 205367 by hardmath last updated on 18/Mar/24 Answered by Ghisom last updated on 18/Mar/24 $$\left(\frac{{n}^{\mathrm{3}/\mathrm{2}} }{\mathrm{3}^{{n}} }+\frac{{n}^{\mathrm{2}} }{\mathrm{4}^{{n}} }+\frac{{n}^{\mathrm{5}/\mathrm{2}} }{\mathrm{5}^{{n}} }\right)^{\mathrm{1}/{n}} = \\…
Question Number 205360 by Davidtim last updated on 18/Mar/24 Answered by A5T last updated on 18/Mar/24 Commented by A5T last updated on 19/Mar/24 $${FE}={EG};{EB}=\sqrt{\mathrm{4}^{\mathrm{2}} +\left(\mathrm{3}−{FE}\right)^{\mathrm{2}}…
Question Number 205361 by Lambertician last updated on 18/Mar/24 Answered by Berbere last updated on 18/Mar/24 $$=−\mathrm{4}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{xln}\left(\frac{\mathrm{1}−{x}}{\mathrm{1}+{x}}\right)}{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}{dx}−\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{1}}{\:\sqrt{\boldsymbol{{x}}\left(\mathrm{1}−\boldsymbol{{x}}\right)}} \\ $$$$=−\mathrm{4}{a}−{q} \\…
Question Number 205353 by MATHEMATICSAM last updated on 17/Mar/24 $$\mathrm{If}\:{ax}^{\mathrm{2}} \:+\:{bx}\:+\:{c}\:=\:\mathrm{0}\:\mathrm{had}\:\mathrm{two}\:\mathrm{roots}\:{p}\:\mathrm{and}\:{q} \\ $$$$\mathrm{and}\:{p}^{\mathrm{2}} \:+\:{q}^{\mathrm{2}} \:=\:{p}^{\mathrm{3}} \:+\:{q}^{\mathrm{3}} \:\mathrm{then}\:\mathrm{show}\:\mathrm{that} \\ $$$${b}^{\mathrm{3}} \:−\:\mathrm{2}{a}^{\mathrm{2}} {c}\:+\:{ab}^{\mathrm{2}} \:=\:\mathrm{3}{abc}. \\ $$ Answered…