Question Number 49661 by maxmathsup by imad last updated on 08/Dec/18 $${calculateA}_{{n}} =\frac{\mathrm{1}}{\mathrm{2}{i}}\:\int_{\mathrm{0}} ^{\mathrm{1}} \left\{\left(\mathrm{1}+{ix}\right)^{{n}} −\left(\mathrm{1}−{ix}\right)^{{n}} \right\}{dx} \\ $$ Commented by maxmathsup by imad last…
Question Number 115193 by mnjuly1970 last updated on 24/Sep/20 $$\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:{mathematics}…\:\: \\ $$$$\:\:\:\:\:\:\:::\:\:\:{digamma}\:\:{limit}\:\::: \\ $$$$\:\:\:\:\:\:\:\:\:\:{if}\:\:\:{k}>\mathrm{0}\:\:{then} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:\:{that}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{lim}_{{x}\rightarrow\mathrm{0}} \:\frac{\mathrm{1}}{{x}}\left(\psi\left(\frac{{k}+{x}}{\mathrm{2}{x}}\right)\:−\:\psi\left(\frac{{k}}{\mathrm{2}{x}}\right)\right)\:=\frac{\mathrm{1}}{{k}}\:\:\:\:\checkmark \\ $$$$ \\ $$$$\:\:\:\:\:{m}.{n}.{july}.\mathrm{1970}……
Question Number 49646 by maxmathsup by imad last updated on 08/Dec/18 $${calculate}\:\int\int_{{D}} \left({x}^{\mathrm{2}} −{y}^{\mathrm{2}} \right)\sqrt{{x}^{\mathrm{2}} \:+{y}^{\mathrm{2}} }{dxdy}\:{with} \\ $$$${D}\:=\left\{\left({x},{y}\right)\in{R}^{\mathrm{2}} /\:\:−\mathrm{1}\leqslant{x}\leqslant\mathrm{1}\:{and}\:\mathrm{0}\leqslant{y}\leqslant\mathrm{2}\:\right\} \\ $$ Terms of Service…
Question Number 49645 by maxmathsup by imad last updated on 08/Dec/18 $${calculate}\:\int\int_{{C}} \:\mid{x}+{y}\mid{dxdy}\:\:{with}\:{C}=\left[−\mathrm{1},\mathrm{1}\right]×\left[−\mathrm{1},\mathrm{1}\right] \\ $$ Answered by ajfour last updated on 08/Dec/18 $${I}\:=\:\mathrm{2}\int_{−\mathrm{1}} ^{\:\:\mathrm{1}} \left[\int_{−{x}}…
Question Number 49636 by maxmathsup by imad last updated on 08/Dec/18 $$\left.\mathrm{1}\right)\:{calculate}\:{A}_{{n}} =\int_{\mathrm{0}} ^{\infty} \:\:{e}^{−{n}\left[{x}\right]} {sin}\left({x}\right){dx}\:{with}\:{n}\:{integr}\:{and}\:{n}\geqslant\mathrm{1} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{nature}\:{of}\:\sum_{{n}=\mathrm{1}} ^{\infty} \:{A}_{{n}} \\ $$ Commented by Abdo…
Question Number 49635 by maxmathsup by imad last updated on 08/Dec/18 $$\left.\mathrm{1}\right){find}\:{f}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{sint}}{\mathrm{2}+{x}\:{cos}\left(\mathrm{2}{t}\right)}{dt} \\ $$$$\left.\mathrm{2}\right)\:{find}\:{g}\left({x}\right)\:=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\frac{{sint}\:{sin}\left(\mathrm{2}{t}\right.}{\left(\mathrm{2}+{x}\:{cos}\left(\mathrm{2}{t}\right)\right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\:\:\frac{{sint}}{\mathrm{2}+\mathrm{3}\:{cos}\left(\mathrm{2}{t}\right)}{dt}\:\:{and}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} \:\frac{{sin}\left({t}\right){sin}\left(\mathrm{2}{t}\right)}{\left(\mathrm{2}+\mathrm{3}{cos}\left(\mathrm{2}{t}\right)\right)^{\mathrm{2}}…
Question Number 115169 by bobhans last updated on 24/Sep/20 $$\int\:\mathrm{sin}\:^{\mathrm{2}} \left(\mathrm{ln}\:{x}\right)\:{dx}\: \\ $$ Answered by bobhans last updated on 24/Sep/20 Answered by Bird last updated…
Question Number 180680 by cortano1 last updated on 15/Nov/22 Answered by ARUNG_Brandon_MBU last updated on 16/Nov/22 $${I}=\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{x}−\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}{{x}+\mathrm{1}+\sqrt{{x}^{\mathrm{2}} +\mathrm{1}}}{dx},\:{x}=\mathrm{sinh}\theta \\ $$$$\:\:\:=\int_{\mathrm{0}} ^{\mathrm{ln}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right)} \frac{\mathrm{sinh}\theta−\mathrm{1}+\mathrm{cosh}\theta}{\mathrm{sinh}\theta+\mathrm{1}+\mathrm{cosh}\theta}\left(\mathrm{cosh}\theta{d}\theta\right)…
Question Number 115111 by mnjuly1970 last updated on 23/Sep/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:…{mathematical}\:\:{analysis}…\:\:\:\:\: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:{prove}\:\:{that}: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left({x}^{\mathrm{8}} +\mathrm{1}\right){ln}\left({x}\right)}{{x}^{\mathrm{10}} −\mathrm{1}}\:{dx}=\frac{\pi^{\mathrm{2}} \varphi^{\mathrm{2}} }{\mathrm{25}}\:\:\checkmark \\…
Question Number 115071 by bemath last updated on 23/Sep/20 $$\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{2}}} {\int}}\:\frac{\mathrm{cos}\:{x}}{\:\sqrt{\mathrm{1}−\mathrm{sin}\:{x}}}\:{dx}\:? \\ $$ Answered by Dwaipayan Shikari last updated on 23/Sep/20 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \frac{{cosx}}{\:\sqrt{\mathrm{1}−{sinx}}}{dx}…