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Category: Differential Equation

set-x-y-z-x-2-y-2-z-2-1-certificate-4-3-2-1-3-x-2y-2z-5-1-3-dv-8-3-

Question Number 211800 by MrGaster last updated on 21/Sep/24 $$ \\ $$$$\boldsymbol{{set}}\:\boldsymbol{\Omega}=\left\{\left(\boldsymbol{{x}},\boldsymbol{{y}},\boldsymbol{{z}}\right)\mid\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{{y}}^{\mathrm{2}} +\boldsymbol{{z}}^{\mathrm{2}} \leq\mathrm{1}\right\}, \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{certificat}}\mathrm{e}: \\ $$$$\:\:\frac{\mathrm{4}\boldsymbol{\pi}}{\mathrm{3}}\sqrt[{\mathrm{3}}]{\mathrm{2}}\leq\int\underset{\boldsymbol{\Omega}} {\int}\int\sqrt[{\mathrm{3}}]{\boldsymbol{{x}}+\mathrm{2}\boldsymbol{{y}}−\mathrm{2}\boldsymbol{{z}}+\mathrm{5}}\boldsymbol{{dv}}\leq\frac{\mathrm{8}\boldsymbol{\pi}}{\mathrm{3}} \\ $$$$ \\ $$$$ \\…

Calculate-the-quadruple-integralas-follows-I-0-1-0-x-0-y-0-z-sin-x-2-y-2-z-2-w-2-1-w-2-z-2-dw-dz-dy-dx-

Question Number 211703 by MrGaster last updated on 17/Sep/24 $$ \\ $$$$\:\mathrm{Calculate}\:\mathrm{the}\:\mathrm{quadruple}\: \\ $$$$\mathrm{integralas}\:\mathrm{follows}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{I}}=\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\boldsymbol{{x}}} \int_{\mathrm{0}} ^{\boldsymbol{\mathrm{y}}} \int_{\mathrm{0}} ^{\boldsymbol{{z}}} \frac{\boldsymbol{\mathrm{sin}}\left(\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{y}}^{\mathrm{2}}…

certificate-I-0-2-x-2-ln-2-cos-x-lncos-x-dx-2-2ln2-

Question Number 211560 by MrGaster last updated on 13/Sep/24 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\mathrm{certificate}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{I}}=\int_{\mathrm{0}} ^{\frac{\boldsymbol{\pi}}{\mathrm{2}}} \sqrt{\sqrt{\boldsymbol{{x}}^{\mathrm{2}} +\boldsymbol{\mathrm{ln}}^{\mathrm{2}} \boldsymbol{\mathrm{cos}}\left(\boldsymbol{{x}}\right)−\boldsymbol{\mathrm{lncos}}\left(\boldsymbol{{x}}\right)\boldsymbol{{dx}}}}=\frac{\boldsymbol{\pi}}{\mathrm{2}}\sqrt{\mathrm{2}\boldsymbol{\mathrm{ln}}\mathrm{2}} \\ $$$$ \\ $$ Terms of Service…

dx-x-2-4x-13-

Question Number 211546 by MrGaster last updated on 12/Sep/24 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\int\frac{\boldsymbol{{dx}}}{\:\sqrt{\boldsymbol{{x}}^{\mathrm{2}} −\mathrm{4}\boldsymbol{{x}}+\mathrm{13}}}=? \\ $$ Answered by Frix last updated on 12/Sep/24 $$\int\frac{{dx}}{\:\sqrt{{x}^{\mathrm{2}} −\mathrm{4}{x}+\mathrm{13}}}\:\overset{\left[{t}=\frac{{x}−\mathrm{2}+\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}+\mathrm{13}}}{\mathrm{3}}\right]}…

Question-210786

Question Number 210786 by zhou0429 last updated on 19/Aug/24 Answered by Berbere last updated on 19/Aug/24 $${x}^{\mathrm{9}} +\mathrm{1}=\underset{{k}=\mathrm{0}} {\overset{\mathrm{8}} {\prod}}\left({x}−{e}^{{i}\pi\left(\frac{\mathrm{1}+\mathrm{2}{k}}{\mathrm{9}}\right)} \right)={p}\left({x}\right) \\ $$$$=\Sigma\int\frac{\mathrm{1}}{\left({x}−{e}^{{i}\pi\left(\frac{\mathrm{1}+\mathrm{2}{k}}{\mathrm{9}}\right)} \right){p}'\left({e}^{{i}\pi\left(\frac{\mathrm{1}+\mathrm{2}{k}}{\mathrm{9}}\right)} \right)}{dx}…