Question Number 222191 by wewji12 last updated on 20/Jun/25 $$\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\:\int_{\:{V}^{\:\mathrm{3}} } \rho_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\mathrm{d}{V}=−\oint_{\:\partial{V}} \:\boldsymbol{\mathrm{J}}_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\centerdot\mathrm{d}\boldsymbol{\mathrm{a}}+\int_{\:{V}^{\:\mathrm{3}} } \:{S}_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\mathrm{d}{V} \\ $$$$\int_{\:{V}^{\:\mathrm{3}} } \:\frac{\partial\rho_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)}{\partial{t}}\:\mathrm{dV}=−\int_{{V}^{\:\mathrm{3}} } \overset{\rightarrow}…
Question Number 222217 by MrGaster last updated on 20/Jun/25 $$\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}\:{x}\:\mathrm{ln}\:{y}}{\:\sqrt{{xy}}}\mathrm{cos}\left({x}+{y}\right)=\pi^{\mathrm{2}} \left(\gamma+\mathrm{2}\:\mathrm{ln}\:\mathrm{2}\right) \\ $$$$\mathrm{Sol}:\int_{\mathrm{0}} ^{\infty} \int_{\mathrm{0}} ^{\infty} \frac{\mathrm{ln}\:{x}\:\mathrm{ln}\:{y}}{\:\sqrt{{xy}}}\mathrm{cos}\left({x}+{y}\right){dxdy}=\mathrm{Re}\left(\left(\int_{\mathrm{0}} ^{\infty} {x}^{−\frac{\mathrm{1}}{\mathrm{2}}} {e}^{{ix}} \mathrm{ln}\:{xdx}\right)\left(\int_{\mathrm{0}}…
Question Number 222218 by Nicholas666 last updated on 20/Jun/25 $$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\int_{\mathrm{0}} ^{\:\pi} \:\mathrm{tan}^{−\mathrm{1}} \:\left(\frac{\mathrm{ln}\:\mathrm{sin}\left({x}\right)}{{x}}\right)\:{dx} \\ $$$$ \\ $$ Answered by Nicholas666 last updated on…
Question Number 222245 by Nicholas666 last updated on 20/Jun/25 $$ \\ $$$$\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} \:\:\frac{\mathrm{cos}^{−\mathrm{1}} \left(\frac{\sqrt{\mathrm{1}\:−\:\mathrm{sin}^{\mathrm{2}} \left({x}\right)\:\mathrm{cos}^{\mathrm{2}} \left({x}\right)}}{\mathrm{1}\:+\:\mathrm{sin}^{\mathrm{2}} \left({x}\right)}\right)\centerdot\mathrm{ln}\left(\frac{\mathrm{1}\:+\:\mathrm{sin}\left({x}\right)}{\mathrm{1}\:+\:\mathrm{cos}\left({x}\right)}\right)}{\:\sqrt{\mathrm{1}\:+\:\mathrm{cos}^{\mathrm{2}} \left({x}\right)\:−\:\mathrm{sin}^{\mathrm{2}} \left({x}\right)}}\:\:\mathrm{d}{x}\:\:\:\:\:\: \\ $$$$ \\ $$ Answered…
Question Number 222172 by hardmath last updated on 19/Jun/25 $$\frac{\mathrm{4}}{\mathrm{5}}\:>\:\frac{\mathrm{8}}{\mathrm{3x}\:−\:\mathrm{6}}\:>\:\frac{\mathrm{2}}{\mathrm{9}} \\ $$$$\mathrm{find}:\:\:\:\boldsymbol{\mathrm{x}}\:=\:? \\ $$ Answered by A5T last updated on 19/Jun/25 $$\frac{\mathrm{4}}{\mathrm{5}}>\frac{\mathrm{8}}{\mathrm{3x}−\mathrm{6}}\:\mathrm{and}\:\mathrm{3x}−\mathrm{6}>\mathrm{0}\:\left(\mathrm{x}>\mathrm{2}\right)\Rightarrow\:\mathrm{12x}−\mathrm{24}>\mathrm{40} \\ $$$$\Rightarrow\mathrm{x}>\frac{\mathrm{16}}{\mathrm{3}}\:…\:\left(\mathrm{i}\right) \\…
Question Number 222141 by fantastic last updated on 19/Jun/25 $${x}^{\mathrm{2}} +\left(\frac{{x}}{\mathrm{2}{x}−\mathrm{1}}\right)^{\mathrm{2}} =\mathrm{12} \\ $$ Answered by Rasheed.Sindhi last updated on 19/Jun/25 $${x}^{\mathrm{2}} \left(\mathrm{2}{x}−\mathrm{1}\right)^{\mathrm{2}} −\mathrm{12}\left(\mathrm{2}{x}−\mathrm{1}\right)^{\mathrm{2}} +{x}^{\mathrm{2}}…
Question Number 222142 by fantastic last updated on 19/Jun/25 $${a}=\mathrm{3}\sqrt{\mathrm{2}}\:,{b}=\frac{\mathrm{1}}{\mathrm{5}^{\frac{\mathrm{1}}{\mathrm{6}}} \sqrt{\mathrm{6}}}\:{and}\:{x},{y}\epsilon\mathbb{R}\:{such}\:{that} \\ $$$$\mathrm{3}{x}\:+\mathrm{2}{y}=\mathrm{log}\:_{{a}} \left(\mathrm{18}\right)^{\frac{\mathrm{5}}{\mathrm{4}}} \\ $$$$\mathrm{2}{x}−{y}=\mathrm{log}\:_{{b}} \left(\sqrt{\mathrm{1080}}\right) \\ $$$${then}\:{find}\:{the}\:{value}\:{of}\:\: \\ $$$$\mathrm{4}{x}+\mathrm{5}{y} \\ $$ Answered by…
Question Number 222175 by klipto last updated on 19/Jun/25 $$\boldsymbol{\mathrm{solve}} \\ $$$$\left(\boldsymbol{\mathrm{e}}^{\mathrm{2}\boldsymbol{\mathrm{y}}} −\boldsymbol{\mathrm{y}}\right)\boldsymbol{\mathrm{cosx}}\frac{\boldsymbol{\mathrm{dy}}}{\boldsymbol{\mathrm{dx}}}=\boldsymbol{\mathrm{e}}^{\boldsymbol{\mathrm{y}}} \boldsymbol{\mathrm{sin}}\mathrm{2}\boldsymbol{\mathrm{x}} \\ $$$$\boldsymbol{\mathrm{klipto}}−\boldsymbol{\mathrm{quanta}} \\ $$ Answered by som(math1967) last updated on 20/Jun/25…
Question Number 222184 by hardmath last updated on 19/Jun/25 $$\mathrm{f}\left(\mathrm{2}\right)\:=\:\mathrm{8} \\ $$$$\mathrm{f}\left(\mathrm{3}\right)\:=\:\mathrm{5} \\ $$$$\int_{\mathrm{2}} ^{\:\mathrm{3}} \:\left(\mathrm{f}\left(\mathrm{x}\right)\:+\:\mathrm{f}\:^{'} \left(\mathrm{x}\right)\centerdot\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$ Answered by wewji12 last updated on…
Question Number 222153 by fantastic last updated on 19/Jun/25 Commented by fantastic last updated on 19/Jun/25 $${area}\:{if}\:{side}\:{length}\:{of}\:{square}\:{is}\:{a} \\ $$ Answered by mr W last updated…