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Author: Tinku Tara

Question-222415

Question Number 222415 by Tawa11 last updated on 25/Jun/25 Answered by MrGaster last updated on 26/Jun/25 $${t}=\sqrt{\frac{\mathrm{1}−{x}}{\mathrm{2}−{x}}} \\ $$$${x}=\frac{\mathrm{1}−\mathrm{2}{t}^{\mathrm{2}} }{\mathrm{1}−{t}^{\mathrm{2}} } \\ $$$${dx}=\frac{−\mathrm{2}{t}}{\left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{\mathrm{2}} }{dt}…

Solve-0-pi-2-ln-n-sin-sin-p-cos-q-d-for-n-p-q-R-0-

Question Number 222409 by Nicholas666 last updated on 25/Jun/25 $$ \\ $$$$\:\:\:\mathrm{Solve}\:;\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{\mathrm{ln}^{{n}} \:\mathrm{sin}\:\theta}{\mathrm{sin}^{{p}} \:\theta\:\mathrm{cos}^{{q}} \:\theta}\:\mathrm{d}\theta\:,\:\mathrm{for}\:{n},{p},{q}\:\in\:\mathbb{R}_{\geqslant\:\mathrm{0}} \:\:\:\:\:\:\:\: \\ $$$$ \\ $$ Terms of Service…

1-0-1-ln-x-ln-1-x-2-ln-1-x-2-1-x-2-dx-2-0-1-ln-x-ln-1-x-ln-1-x-ln-1-x-2-1-x-dx-3-0-1-ln-x-ln-1-x-2-ln-1-x-2-x-dx

Question Number 222411 by Nicholas666 last updated on 25/Jun/25 $$\:\: \\ $$$$\:\:\:\left[\:\mathrm{1}\:.\right]\:\:\:\:\int_{\mathrm{0}} ^{\:\mathrm{1}} \:\frac{\mathrm{ln}\left({x}\right)\:\mathrm{ln}\left(\mathrm{1}−{x}^{\mathrm{2}} \right)\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}−{x}^{\mathrm{2}} }\:\:\:{dx}\:\:\:\:\: \\ $$$$\:\:\:\left[\:\mathrm{2}\:.\right]\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{ln}\left({x}\right)\:\mathrm{ln}\left(\mathrm{1}−{x}\right)\:\mathrm{ln}\left(\mathrm{1}+{x}\right)\:\mathrm{ln}\left(\mathrm{1}+{x}^{\mathrm{2}} \right)}{\mathrm{1}+{x}}\:{dx} \\ $$$$\:\left[\:\mathrm{3}\:.\right]\:\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}}…

Prove-S-E-dS-enc-0-E-r-r-3-

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…