Question Number 123745 by mnjuly1970 last updated on 27/Nov/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{advanced}\:\:\:{calculus}… \\ $$$$\:\:{evaluation}\:\:{of}: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{2}}} {sin}\left({x}\right){log}\left({sin}\left({x}\right)\right){dx} \\ $$$$\:\:\:\:{by}\:{using}\:{the}\:{euler}\:{beta}\:{and}\:{gamma}\:{function}: \\ $$$$\:\:\:\:\:\:\:\:\beta\left({p},\frac{\mathrm{1}}{\mathrm{2}}\right)=\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} {sin}^{\mathrm{2}{p}−\mathrm{1}} \left({x}\right){dx} \\ $$$$\:\:\:\frac{{d}\beta\left({p},\frac{\mathrm{1}}{\mathrm{2}}\right)}{{dp}}\:=\mathrm{2}\int_{\mathrm{0}}…
Question Number 189266 by cortano12 last updated on 14/Mar/23 $$\:\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\sqrt[{\mathrm{3}}]{\mathrm{tan}\:\mathrm{x}}}{\mathrm{1}+\mathrm{sin}\:\mathrm{2x}}\:\mathrm{dx}\:=? \\ $$ Answered by MJS_new last updated on 14/Mar/23 $$\int\frac{\sqrt[{\mathrm{3}}]{\mathrm{tan}\:{x}}}{\mathrm{1}+\mathrm{sin}\:\mathrm{2}{x}}{dx}= \\ $$$$\:\:\:\:\:\left[{t}=\sqrt[{\mathrm{3}}]{\mathrm{tan}\:{x}}\:\rightarrow\:{dx}=\mathrm{3cos}^{\mathrm{2}} \:{x}\:\sqrt[{\mathrm{3}}]{\mathrm{tan}^{\mathrm{2}}…
Question Number 58187 by maxmathsup by imad last updated on 19/Apr/19 $${let}\:{f}\left({x}\right)\:=\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left({x}+\frac{{x}}{{t}}\right){dt}\:\:\:{withx}>\mathrm{0} \\ $$$$\left.\mathrm{1}\right)\:{determine}\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:\:{give}\:{f}^{'} \left({x}\right)\:{at}\:{form}\:{of}\:{integral}\:{and}\:{find}\:{its}\:{value} \\ $$$$\left.\mathrm{3}\right)\:{calculate}\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left(\mathrm{1}+\frac{\mathrm{1}}{{t}}\right){dt}\:\:\:{and}\:\:\:\:\int_{\mathrm{1}} ^{\mathrm{3}} \:{arctan}\left(\mathrm{2}+\frac{\mathrm{2}}{{t}}\right){dt}\:.…
Question Number 58185 by maxmathsup by imad last updated on 19/Apr/19 $${find}\:\int\:\:\:\frac{{xdx}}{{cosx}\:+{sin}\left(\mathrm{2}{x}\right)} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 58184 by maxmathsup by imad last updated on 19/Apr/19 $${find}\:\:\int\:\:\:\:\:\:\frac{{xdx}}{{sinx}\:+{cos}\left(\mathrm{2}{x}\right)} \\ $$ Terms of Service Privacy Policy Contact: info@tinkutara.com
Question Number 189250 by sonukgindia last updated on 13/Mar/23 Commented by MJS_new last updated on 13/Mar/23 $$\mathrm{we}\:\mathrm{can}\:\mathrm{only}\:\mathrm{approximate} \\ $$ Terms of Service Privacy Policy Contact:…
Question Number 123710 by Bird last updated on 27/Nov/20 $${calculate}\:\:\int_{\mathrm{1}} ^{\sqrt{\mathrm{3}}} \:\:\:\:\frac{{dx}}{\left({x}^{\mathrm{2}} +\mathrm{1}\right)^{\mathrm{2}} \left({x}+\mathrm{2}\right)^{\mathrm{5}} } \\ $$ Answered by MJS_new last updated on 27/Nov/20 $$\mathrm{Ostrogradski}\:\mathrm{gives}…
Question Number 123703 by Algoritm last updated on 27/Nov/20 Answered by MJS_new last updated on 27/Nov/20 $$=\int\left(\mathrm{cos}\:{x}\right)^{\mathrm{5}/\mathrm{3}} {dx} \\ $$$$\mathrm{use}\:\mathrm{these}\:\mathrm{formulas}: \\ $$$$\int\left(\mathrm{sin}\:{x}\right)^{{p}/{q}} {dx}=\frac{{q}}{{p}+{q}}\left(\mathrm{sin}\:{x}\right)^{\frac{{p}+{q}}{{q}}} \:_{\mathrm{2}} \mathrm{F}_{\mathrm{1}}…
Question Number 58168 by maxmathsup by imad last updated on 19/Apr/19 $${find}\:\int\:\:\:\:\frac{\sqrt{{tanx}}}{{sin}\left(\mathrm{2}{x}\right)}{dx} \\ $$ Answered by tanmay last updated on 19/Apr/19 $$\int\frac{\sqrt{{tanx}}\:}{\mathrm{2}{tanx}}×\left(\mathrm{1}+{tan}^{\mathrm{2}} {x}\right){dx} \\ $$$$\frac{\mathrm{1}}{\mathrm{2}}\int\frac{{sec}^{\mathrm{2}}…
Question Number 123687 by mnjuly1970 last updated on 27/Nov/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:…{nice}\:\:{calculus}.. \\ $$$$\:{prove}\:{that}::\:\:\:\:{lim}_{{x}\rightarrow\mathrm{0}} \left(\frac{\mathrm{2}\phi\left({x}\right)}{{x}^{\mathrm{2}} }\:+\frac{\pi^{\mathrm{2}} }{\mathrm{3}{x}}\right)\:\overset{???} {=}\zeta\left(\mathrm{3}\right) \\ $$$$\:\:\:\:\:\:\:\:\:{where} \\ $$$$\:\:\:\:\:\phi\left({x}\right)=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\left({t}^{{x}} −\mathrm{1}\right)\left({ln}\left(\mathrm{1}−{t}\right)\right)}{{tln}\left({t}\right)}{dt} \\ $$…