Question Number 110197 by redmiiuser last updated on 27/Aug/20 $$\left(−\mathrm{1}\right)^{\pi} =? \\ $$$$\left(−\mathrm{1}\right)^{\frac{\mathrm{22}}{\mathrm{7}}} \\ $$$$=\left(−\mathrm{1}\right)^{\mathrm{3}+\frac{\mathrm{1}}{\mathrm{7}}} \\ $$$$=\left(−\mathrm{1}\right)^{\mathrm{3}} .\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{7}}} \\ $$$$=−\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{7}}} \\ $$$${let}\:\left(−\mathrm{1}\right)^{\pi} ={t} \\ $$$$\Rightarrow−\left(−\mathrm{1}\right)^{\frac{\mathrm{1}}{\mathrm{7}}}…
Question Number 110175 by mathdave last updated on 27/Aug/20 $${solve}\:{the}\:{integral} \\ $$$$\int_{−\infty} ^{+\infty} \frac{\mathrm{sin}{x}}{\:\sqrt[{\mathrm{3}}]{{x}}}{dx} \\ $$ Answered by mathmax by abdo last updated on 28/Aug/20…
Question Number 175706 by daus last updated on 05/Sep/22 Commented by daus last updated on 05/Sep/22 $${just}\:{help}\:{me}\:{in}\:\:{j},{k},{l}\: \\ $$ Answered by mahdipoor last updated on…
Question Number 44622 by mondodotto@gmail.com last updated on 02/Oct/18 $$\boldsymbol{\mathrm{if}}\:\boldsymbol{{y}}=\boldsymbol{\mathrm{ln}}\left[\boldsymbol{\mathrm{tan}}\left(\frac{\boldsymbol{\pi}}{\mathrm{4}}+\frac{\boldsymbol{{x}}}{\mathrm{2}}\right)\right]\:\boldsymbol{\mathrm{show}}\:\boldsymbol{\mathrm{that}} \\ $$$$\frac{\boldsymbol{{dy}}}{\boldsymbol{{dx}}}=\boldsymbol{\mathrm{sec}{x}} \\ $$ Answered by tanmay.chaudhury50@gmail.com last updated on 02/Oct/18 $$\frac{{dy}}{{dx}}=\frac{\mathrm{1}}{{tan}\left(\frac{\pi}{\mathrm{4}}+\frac{{x}}{\mathrm{2}}\right)}×{sec}^{\mathrm{2}} \left(\frac{\pi}{\mathrm{4}}+\frac{{x}}{\mathrm{2}}\right)×\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\frac{{dy}}{{dx}}=\frac{\mathrm{1}}{\mathrm{2}}×\frac{{cos}\left(\frac{\pi}{\mathrm{4}}+\frac{{x}}{\mathrm{2}}\right)}{{sin}\left(\frac{\pi}{\mathrm{4}}+\frac{{x}}{\mathrm{2}}\right)}×\frac{\mathrm{1}}{{cos}^{\mathrm{2}}…
Question Number 44623 by mondodotto@gmail.com last updated on 02/Oct/18 $$\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}}…