Question Number 44623 by mondodotto@gmail.com last updated on 02/Oct/18 $$\boldsymbol{\mathrm{given}}\:\boldsymbol{\mathrm{that}}\:\boldsymbol{\mathrm{sin}}^{−\mathrm{1}} \boldsymbol{{x}}+\boldsymbol{\mathrm{sin}}^{−\mathrm{1}} \boldsymbol{{y}}=\boldsymbol{\mathrm{c}} \\ $$$$\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}}\:\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}+\sqrt{\frac{\mathrm{1}−\boldsymbol{{y}}^{\mathrm{2}} }{\mathrm{1}−\boldsymbol{{x}}^{\mathrm{2}} }}=\mathrm{0} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 02/Oct/18…
Question Number 110154 by mathdave last updated on 27/Aug/20 $${prove}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \int_{\mathrm{0}} ^{\mathrm{1}} \frac{\mathrm{tanh}^{−\mathrm{1}} \left(^{\mathrm{4}} \sqrt{{x}}\right)\mathrm{tanh}^{−\mathrm{1}} \left(^{\mathrm{4}} \sqrt{{y}}\right)}{{x}\sqrt{{y}}}=\pi^{\mathrm{2}} \\ $$ Terms of Service…
Question Number 110149 by mohammad17 last updated on 27/Aug/20 $${find}\:{the}\:{domain}\:{f}\left({x},{y}\right)={x}+\mathrm{4}\sqrt{{y}\:}\:? \\ $$ Commented by mohammad17 last updated on 28/Aug/20 $${are}\:{you}\:{can}\:{help}\:{me} \\ $$ Terms of Service…
Question Number 110145 by mathdave last updated on 27/Aug/20 $${prove}\:{that} \\ $$$$\int_{\mathrm{1}} ^{\infty} \frac{\mathrm{ln}\left({x}\right)}{\mathrm{1}+{x}+{x}^{\mathrm{2}} +{x}^{\mathrm{3}} }{dx}=\frac{{G}}{\mathrm{2}}−\frac{\pi^{\mathrm{2}} }{\mathrm{32}} \\ $$$${G}\left({catalan}\:{constant}\right) \\ $$ Answered by mnjuly1970 last…
Question Number 110118 by mathdave last updated on 27/Aug/20 $${prove}\:{that} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} \Gamma\left(\mathrm{1}−\frac{{x}}{\mathrm{2}}\right)\Gamma\left(\mathrm{1}+\frac{{x}}{\mathrm{2}}\right){dx}=\frac{\mathrm{4}}{\pi}{G} \\ $$$${where}\:{G}\left({catalan}\:{constant}\right) \\ $$ Commented by Sarah85 last updated on 27/Aug/20…
Question Number 110112 by mathdave last updated on 27/Aug/20 $${show}\:{that}\: \\ $$$$\int_{\mathrm{0}} ^{\infty} {x}\mathrm{sin}\left({x}^{\mathrm{3}} \right){dx}=\frac{\mathrm{1}}{\mathrm{3}}\bullet\frac{\pi}{\Gamma\left(\frac{\mathrm{1}}{\mathrm{3}}\right)} \\ $$ Answered by mnjuly1970 last updated on 27/Aug/20 $${x}^{\mathrm{3}}…
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Question Number 110109 by aurpeyz last updated on 27/Aug/20 Answered by $@y@m last updated on 27/Aug/20 Commented by Rio Michael last updated on 27/Aug/20 $$\left(\mathrm{3}{x}\right)^{−\mathrm{2}}…
Question Number 110096 by mathdave last updated on 27/Aug/20 $${find}\:{the}\:{following}\:{product}\:{integral} \\ $$$$\:\:\:\:\:\:\:\:\left(\mathrm{1}\right)\:\:\:\:\int\left({x}\right)^{{dx}} \\ $$$$\:\:\:\:\:\:\:\:\:\left(\mathrm{2}\right)\:\:\:\:\int\left({e}^{{x}} \right)^{{dx}} \\ $$ Commented by Her_Majesty last updated on 27/Aug/20 $${if}\:{you}\:{use}\:{the}\:{same}\:{definition}\:{of}\:{product}…
Question Number 110092 by bemath last updated on 27/Aug/20 Answered by 1549442205PVT last updated on 27/Aug/20 $$\left.\mathrm{30}\right)\mathrm{Since}\:\mathrm{AK}=\mathrm{KL}=\mathrm{LB}=\mathrm{m}\left(\mathrm{1}\right),\mathrm{BA}=\mathrm{BC}=\mathrm{a} \\ $$$$\left(\mathrm{hypothesis}\right)\Rightarrow\mathrm{LC}=\mathrm{BC}−\mathrm{LB}=\mathrm{a}−\mathrm{m}\left(\mathrm{2}\right) \\ $$$$\mathrm{but} \\ $$$$\mathrm{a}−\mathrm{m}=\mathrm{AB}−\mathrm{AK}=\mathrm{KB}=\mathrm{AC}\left(\mathrm{3}\right)\left(\mathrm{hypothesis}\right)? \\ $$$$\mathrm{From}\left(\mathrm{2}\right)\left(\mathrm{3}\right)\mathrm{we}\:\mathrm{infer}\:\mathrm{LC}=\mathrm{KB}=\mathrm{AC}\left(\mathrm{4}\right)…