Question Number 222365 by wewji12 last updated on 23/Jun/25 $$\mathrm{Prove} \\ $$$$\oint_{\:\partial\mathcal{S}} \:\overset{\rightarrow} {\boldsymbol{\mathrm{E}}}\centerdot\mathrm{d}\overset{\rightarrow} {\mathcal{S}}=\frac{\rho_{\mathrm{enc}} }{\boldsymbol{\varepsilon}_{\mathrm{0}} } \\ $$$$\overset{\rightarrow} {\boldsymbol{\mathrm{E}}}=\frac{\overset{\rightarrow} {\boldsymbol{\mathrm{r}}}}{\boldsymbol{\mathrm{r}}^{\mathrm{3}} }\: \\ $$ Answered…
Question Number 222356 by wewji12 last updated on 23/Jun/25 $$\int_{\mathrm{0}} ^{\:\infty} \:{f}\left({r}\right)\mathrm{d}{r}=\mathrm{1}\:,\:\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{g}\left({r}\right)\mathrm{d}{r}=\mathrm{1} \\ $$$$\int_{−\infty\boldsymbol{{i}}+\boldsymbol{\gamma}} ^{\:\:\infty\boldsymbol{{i}}+\boldsymbol{\gamma}} \:{F}\left({t}\right){G}\left({t}\right)\mathrm{d}{t}=?? \\ $$$${F}\left({t}\right)=\int_{\mathrm{0}} ^{\:\infty} \:{f}\left({r}\right){e}^{−{rt}} \mathrm{d}{r}\:,\:{G}\left({t}\right)=\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{g}\left({r}\right){e}^{−{rt}}…
Question Number 222300 by fantastic last updated on 22/Jun/25 $${How}\:{do}\:{you}\:{put}\:{a}\:{box}\:{around}\:{something}??\:{please}\:{tell}\:{me} \\ $$ Commented by mr W last updated on 22/Jun/25 $$\begin{array}{|c|}{{something}\:{like}\:{this}?}\\\hline\end{array} \\ $$ Commented by…
Question Number 222317 by wewji12 last updated on 22/Jun/25 $$\int_{\mathrm{0}} ^{\:\infty} \:{f}\left({z}\right)\mathrm{d}{z}=\frac{\pi}{\mathrm{2}}\:,\:\int_{\mathrm{0}} ^{\:\infty} \:\mathrm{g}\left({z}\right)\mathrm{d}{z}=\mathrm{1} \\ $$$$\frac{\mathrm{2}}{\pi}\int_{\mathrm{0}} ^{\:\infty} \:{f}\left({z}\right)\mathrm{g}\left({z}\right)\mathrm{d}{z}=?? \\ $$ Answered by MrGaster last updated…
Question Number 222280 by klipto last updated on 21/Jun/25 $$\boldsymbol{\mathrm{y}}=\mathrm{3}\boldsymbol{\mathrm{x}}^{\mathrm{2024}} −\mathrm{18}\boldsymbol{\mathrm{x}}^{\mathrm{2020}} +\mathrm{5}\boldsymbol{\mathrm{x}}^{\mathrm{47}} −\mathrm{8} \\ $$$$\boldsymbol{\mathrm{find}}\:\frac{\boldsymbol{\mathrm{d}}^{\mathrm{2025}} \boldsymbol{\mathrm{y}}}{\boldsymbol{\mathrm{dx}}^{\mathrm{2025}} } \\ $$ Commented by mr W last updated…
Question Number 222193 by MrGaster last updated on 20/Jun/25 $$\mathrm{Prove}: \\ $$$$\frac{\mathrm{1}}{\mathrm{sin}\left(\frac{\mathrm{3}\pi}{\mathrm{8}}\right)}=\mathrm{2}\left(\mathrm{cos}\left(\frac{\pi}{\mathrm{8}}\right)−\mathrm{sin}\left(\frac{\pi}{\mathrm{8}}\right)\right) \\ $$$$\frac{\mathrm{2}}{\mathrm{sin}\left(\frac{\mathrm{8}\pi}{\mathrm{9}}\right)}−\frac{\mathrm{1}}{\mathrm{sin}\left(\frac{\mathrm{5}\pi}{\mathrm{9}}\right)}=\mathrm{2}\sqrt{\mathrm{3}}+\mathrm{4sin}\left(\frac{\pi}{\mathrm{9}}\right) \\ $$$$\frac{\mathrm{2}}{\mathrm{sin}\left(\frac{\mathrm{6}\pi}{\mathrm{7}}\right)}+\frac{\mathrm{1}}{\mathrm{sin}\left(\frac{\mathrm{4}\pi}{\mathrm{7}}\right)}=\mathrm{4}\left(\mathrm{sin}\left(\frac{\pi}{\mathrm{7}}\right)+\mathrm{cos}\left(\frac{\pi}{\mathrm{14}}\right)\right) \\ $$ Answered by som(math1967) last updated on 20/Jun/25…
Question Number 222191 by wewji12 last updated on 20/Jun/25 $$\frac{\mathrm{d}\:\:}{\mathrm{d}{t}}\:\int_{\:{V}^{\:\mathrm{3}} } \rho_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\mathrm{d}{V}=−\oint_{\:\partial{V}} \:\boldsymbol{\mathrm{J}}_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\centerdot\mathrm{d}\boldsymbol{\mathrm{a}}+\int_{\:{V}^{\:\mathrm{3}} } \:{S}_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)\mathrm{d}{V} \\ $$$$\int_{\:{V}^{\:\mathrm{3}} } \:\frac{\partial\rho_{{q}} \left(\boldsymbol{\mathrm{r}},{t}\right)}{\partial{t}}\:\mathrm{dV}=−\int_{{V}^{\:\mathrm{3}} } \overset{\rightarrow}…
Question Number 222142 by fantastic last updated on 19/Jun/25 $${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 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 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}}…