Question Number 211987 by Spillover last updated on 26/Sep/24 Commented by MathematicalUser2357 last updated on 26/Sep/24 $$\mathrm{No}\:\mathrm{closed}\:\mathrm{forms}\:\mathrm{for}\:\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {x}^{\mathrm{2}} \left(\frac{\mathrm{1}}{\:\sqrt{\mathrm{tan}\:{x}}}−\sqrt{\mathrm{tan}\:{x}}\right){dx}. \\ $$$$\mathrm{But}\:\mathrm{you}\:\mathrm{can}\:\mathrm{just}\:\mathrm{approximate}. \\ $$ Commented…
Question Number 212011 by a.lgnaoui last updated on 28/Sep/24 $$\boldsymbol{\mathrm{D}}\mathrm{eterminer}:\:\:\boldsymbol{\mathrm{R}}\mathrm{1}\:\:\:\boldsymbol{\mathrm{R}}\mathrm{2}\:\:\:\boldsymbol{\mathrm{R}}\mathrm{3} \\ $$$$\boldsymbol{\mathrm{pour}}\:\:\:\boldsymbol{\mathrm{b}}=\mathrm{12}\boldsymbol{\mathrm{cm}}\:\:\:\: \\ $$$$\boldsymbol{\mathrm{EF}}\://\:\boldsymbol{\mathrm{MN}}\:;\:\:\boldsymbol{\mathrm{EF}}\:\boldsymbol{\mathrm{Tangent}}\:\boldsymbol{\mathrm{aux}}\:\boldsymbol{\mathrm{cercles}}:\: \\ $$$$\:\:\boldsymbol{\mathrm{C}}\mathrm{1}\left(\boldsymbol{\mathrm{R}}\mathrm{1}\right)\:\:\:\boldsymbol{\mathrm{C}}\mathrm{2}\left(\boldsymbol{\mathrm{R}}\mathrm{2}\right)\:\:;\:\:\:\boldsymbol{\mathrm{EF}}=\boldsymbol{\mathrm{a}}\:\:\:\:\:\:\boldsymbol{\mathrm{MN}}=\boldsymbol{\mathrm{b}} \\ $$$$\boldsymbol{\mathrm{MN}}:\:\boldsymbol{\mathrm{tangent}}\:\boldsymbol{\mathrm{au}}\:\boldsymbol{\mathrm{cercle}}\:\boldsymbol{\mathrm{C}}\mathrm{2} \\ $$$$\boldsymbol{\mathrm{OM}}=\boldsymbol{\mathrm{ON}}=\frac{\mathrm{3}\boldsymbol{\mathrm{a}}}{\mathrm{2}}\:\:\:\:\:\:\:\:\measuredangle\mathrm{MON}=\mathrm{2}\boldsymbol{\mathrm{x}}\:\:\: \\ $$$$ \\ $$ Commented…
Question Number 212006 by RojaTaniya last updated on 26/Sep/24 Answered by a.lgnaoui last updated on 26/Sep/24 $$\:\:\mathrm{posons}\:\:\boldsymbol{\mathrm{z}}=\boldsymbol{\mathrm{x}}+\mathrm{100} \\ $$$$\:\:\frac{\left(\boldsymbol{\mathrm{z}}−\mathrm{2}\right)^{\mathrm{5}} +\left(\boldsymbol{\mathrm{z}}+\mathrm{2}\right)^{\mathrm{5}} }{\left(\boldsymbol{\mathrm{z}}−\mathrm{1}\right)^{\mathrm{5}} +\left(\boldsymbol{\mathrm{z}}+\mathrm{1}\right)^{\mathrm{5}} }=\frac{\mathrm{2}\boldsymbol{\mathrm{z}}^{\mathrm{5}} +\mathrm{80}\boldsymbol{\mathrm{z}}^{\mathrm{3}} +\mathrm{160}\boldsymbol{\mathrm{z}}}{\mathrm{2}\boldsymbol{\mathrm{z}}^{\mathrm{5}}…
Question Number 212007 by Nadirhashim last updated on 26/Sep/24 $$\:\:\:\:\int\boldsymbol{{sin}}\left(\boldsymbol{{x}}\right)\:\sqrt[{\mathrm{3}}]{\boldsymbol{{tan}}\left(\boldsymbol{{x}}\right)}\:.\boldsymbol{{dx}} \\ $$ Answered by Ghisom last updated on 26/Sep/24 $$=\int\left(\mathrm{cos}\:{x}\right)^{−\mathrm{1}/\mathrm{3}} \left(\mathrm{sin}\:{x}\right)^{\mathrm{4}/\mathrm{3}} {dx}= \\ $$$$=\frac{\mathrm{3}}{\mathrm{7}}\:_{\mathrm{2}} {F}_{\mathrm{1}}…
Question Number 212001 by mnjuly1970 last updated on 26/Sep/24 $$ \\ $$$$\:\:\:\:{prove}\:\:{that}: \\ $$$$ \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\underset{{k}\in\mathbb{Z}} {\sum}\:\frac{\:\left(−\mathrm{1}\right)^{{k}} }{\:{x}\:+\:{k}\pi}\:=\:\frac{\mathrm{1}}{{sin}\left({x}\right)}\:\:\: \\ $$$$\:\:\:\:\:\:\:\:−−−−−−−−− \\ $$ Answered…
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Question Number 211949 by Spillover last updated on 25/Sep/24 Answered by MathematicalUser2357 last updated on 26/Sep/24 $$\mathrm{Thanks}\:\mathrm{for}\:\mathrm{the}\:\mathrm{integration}\:\mathrm{idea}! \\ $$$$\int_{−\mathrm{1}} ^{\mathrm{1}} \frac{\mathrm{1}}{{x}}\sqrt{\frac{\mathrm{1}+{x}}{\mathrm{1}−{x}}}\mathrm{ln}\left(\frac{{ax}^{\mathrm{2}} +\sqrt{\mathrm{2}{a}}{x}+\mathrm{1}}{{ax}^{\mathrm{2}} −\sqrt{\mathrm{2}{a}}{x}+\mathrm{1}}\right){dx}=\mathrm{4}\pi\:\mathrm{cot}^{−\mathrm{1}} \sqrt{\frac{\sqrt{{a}^{\mathrm{2}} +\mathrm{1}}+\mathrm{1}}{{a}}}…
Question Number 211944 by liuxinnan last updated on 25/Sep/24 $$ \\ $$$${f}\left({x}\right)=\frac{\sqrt{\mathrm{1}+{x}}−\sqrt{\mathrm{1}−{x}}}{\:\sqrt{\mathrm{1}+{x}}+\sqrt{\mathrm{1}−{x}}}\:\:\:\:{f}^{'} \left({x}\right)=? \\ $$$$ \\ $$$$ \\ $$ Answered by efronzo1 last updated on…
Question Number 211961 by Nadirhashim last updated on 25/Sep/24 $$\:\:\:\boldsymbol{{if}}\:\:\:\mathrm{7}^{\boldsymbol{{sin}}^{\mathrm{2}\:} \boldsymbol{{x}}} +\:\mathrm{7}^{\boldsymbol{{cos}}^{\mathrm{2}} \boldsymbol{{x}}} =\:\mathrm{8}\:\boldsymbol{{find}}\:\boldsymbol{{x}} \\ $$ Answered by efronzo1 last updated on 25/Sep/24 $$\:\:\mathrm{7}^{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}}…
Question Number 211946 by BaliramKumar last updated on 25/Sep/24 Answered by BHOOPENDRA last updated on 25/Sep/24 $$\left(\mathrm{2}+\mathrm{3}+\mathrm{4}\right)^{\mathrm{2}} =\mathrm{9}^{\mathrm{2}} =\mathrm{81} \\ $$ Commented by BHOOPENDRA last…