Question Number 66332 by mathmax by abdo last updated on 12/Aug/19 $${let}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \sqrt{\mathrm{1}−{x}}{dx} \\ $$$$\left.\mathrm{1}\right){calculate}\:{A}_{\mathrm{0}} \:{and}\:{A}_{\mathrm{1}} \\ $$$$\left.\mathrm{2}\right){prove}\:{that}\:\forall{n}\in{N}^{\bigstar} \:\:\:\:\left(\mathrm{3}+\mathrm{2}{n}\right){A}_{{n}} =\mathrm{2}{nA}_{{n}−\mathrm{1}} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{A}_{{n}}…
Question Number 66335 by mathmax by abdo last updated on 12/Aug/19 $${find}\:\:\int_{−\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{6}}} \:\frac{\mathrm{1}+{tanx}}{\mathrm{1}+{sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$ Commented by Prithwish sen last updated on 13/Aug/19 $$\int_{−\frac{\pi}{\mathrm{6}}}…
Question Number 66330 by mathmax by abdo last updated on 12/Aug/19 $${let}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \:{x}^{{n}} \:{e}^{−{x}} \:{dx}\:\:\:\:{with}\:{n}\:{integr}\:{natural} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}_{\mathrm{0}} \:,\:{I}_{\mathrm{1}} \:{and}\:{I}_{\mathrm{2}} \\ $$$$\left.\mathrm{2}\right){find}\:{arelation}\:{between}\:{I}_{{n}} \:{and}\:{I}_{{n}} \\…
Question Number 66328 by mathmax by abdo last updated on 12/Aug/19 $${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\frac{{dt}}{\left(\mathrm{1}+{t}^{\mathrm{2}} \right)^{\mathrm{3}} } \\ $$ Commented by mathmax by abdo last updated…
Question Number 66326 by mathmax by abdo last updated on 12/Aug/19 $${calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\frac{{x}^{\mathrm{4}} \:+\mathrm{1}}{{x}^{\mathrm{6}} +\mathrm{1}}{dx} \\ $$ Commented by Prithwish sen last updated on…
Question Number 131866 by mnjuly1970 last updated on 09/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:\:{calculus}… \\ $$$$\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\infty} \frac{{dx}}{{x}^{\mathrm{5}} \left({e}^{\frac{\mathrm{1}}{{x}}} −\mathrm{1}\right)}=? \\ $$$$ \\ $$ Answered by mnjuly1970 last updated…
Question Number 66325 by mathmax by abdo last updated on 12/Aug/19 $${calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:{cos}^{\mathrm{4}} {x}\:{sin}^{\mathrm{2}} {x}\:{dx} \\ $$ Commented by Prithwish sen last updated on…
Question Number 789 by 123456 last updated on 14/Mar/15 $$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\underset{{x}−{x}^{\mathrm{2}} } {\overset{\sqrt{{x}−{x}^{\mathrm{2}} }} {\int}}\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{dydx}=? \\ $$$$\int\underset{\mathrm{B}} {\int}\sqrt{{x}^{\mathrm{2}} +{y}^{\mathrm{2}} }{dxdy}\:\:\:\:\:\mathrm{B}=\left\{\left({x},{y}\right)\in\mathbb{R}^{\mathrm{2}} :{y}\geqslant{x}−{x}^{\mathrm{2}} \wedge{x}^{\mathrm{2}}…
Question Number 787 by 123456 last updated on 22/Mar/15 $$\underset{{r}=\mathrm{1}} {\overset{+\infty} {\sum}}\sqrt{\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\mathrm{5}+\Gamma\left(\frac{\mathrm{2}}{\mathrm{2}+{x}^{\mathrm{2}{r}} }\right){dx}} \\ $$ Commented by 123456 last updated on 13/Mar/15 $$\underset{{r}=\mathrm{1}}…
Question Number 131852 by mnjuly1970 last updated on 09/Feb/21 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:\:{analysis}\:\left({II}\right)… \\ $$$$\:\:\:\:{evaluate}\::: \\ $$$$\:\:\:\:\:\:\:\:\varnothing=\int_{\mathrm{1}} ^{\:\mathrm{10}} {x}^{\mathrm{2}} {d}\left(\left\{{x}\right\}\right)=? \\ $$$$\:\:\:\:\:\:\:\left\{{x}\right\}\:::\:{fractional}\:{part}\:{of}\:{x}\:… \\ $$$$ \\ $$ Answered by…