Question Number 125336 by mnjuly1970 last updated on 10/Dec/20 $$\:\:\:\:\:\:\:\:\:\:\:\:\:\:…\:{nice}\:\:{calculus}\:… \\ $$$$\:\:{evaluate}\:: \\ $$$$\:\:\:\phi=\int_{\mathrm{0}\:} ^{\:\mathrm{1}} {x}^{\mathrm{2}} {ln}\left({x}\right).{ln}\left(\mathrm{1}−{x}\right){dx}\:=? \\ $$$$ \\ $$ Answered by Dwaipayan Shikari…
Question Number 59800 by necx1 last updated on 14/May/19 $${find}\:{the}\:{general}\:{solution}\:{y}\left({t}\right)\:{of}\:{the} \\ $$$${ordinary}\:{differential}\:{equation} \\ $$$${y}''\:+\:\omega^{\mathrm{2}} {y}=\mathrm{cos}\:\omega{t}\:,{where}\:{w}>\mathrm{0} \\ $$ Answered by MJS last updated on 15/May/19 $$\mathrm{1}^{\mathrm{st}}…
Question Number 125313 by john_santu last updated on 10/Dec/20 $$\:\:\:\beta\left({x}\right)=\int\:\frac{{x}^{\mathrm{3}} }{\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }}\:{dx}\: \\ $$ Answered by john_santu last updated on 10/Dec/20 $${let}\:\sqrt{\mathrm{1}−{x}^{\mathrm{2}} \:}\:=\:{w}\:\Rightarrow{x}^{\mathrm{2}} \:=\:\mathrm{1}−{w}^{\mathrm{2}} \\…
Question Number 190841 by Rupesh123 last updated on 12/Apr/23 Answered by ARUNG_Brandon_MBU last updated on 12/Apr/23 $${I}=\int_{\mathrm{0}} ^{\pi} \frac{{xdx}}{\mathrm{1}+\mathrm{cos}\alpha\mathrm{sin}{x}}=\int_{\mathrm{0}} ^{\pi} \frac{\pi−{x}}{\mathrm{1}+\mathrm{cos}\alpha\mathrm{sin}{x}}{dx} \\ $$$$\:\:\:=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi} \frac{\pi}{\mathrm{1}+\mathrm{cos}\alpha\mathrm{sin}{x}}{dx}=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}}…
Question Number 190812 by Mahliyo last updated on 12/Apr/23 Commented by Frix last updated on 12/Apr/23 $$\mathrm{Use}\:\mathrm{3}\:\mathrm{steps} \\ $$$$\mathrm{1}.\:{t}={x}+\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{2}.\:{u}=\mathrm{sin}^{−\mathrm{1}} \:\frac{\mathrm{2}{t}}{\:\sqrt{\mathrm{5}}} \\ $$$$\mathrm{3}.\:{v}=\mathrm{tan}\:\frac{{u}}{\mathrm{2}} \\…
Question Number 125276 by Study last updated on 09/Dec/20 $$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\frac{{x}^{\mathrm{9}} −\mathrm{1}}{{lnx}}{dx}=??? \\ $$ Answered by Dwaipayan Shikari last updated on 09/Dec/20 $${I}\left({a}\right)=\int_{\mathrm{0}} ^{\mathrm{1}}…
Question Number 190809 by safojontoshtemirov last updated on 12/Apr/23 Commented by Frix last updated on 12/Apr/23 $$\mathrm{Use}\:\mathrm{2}\:\mathrm{steps}: \\ $$$$\mathrm{1}.\:{t}={x}+\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\mathrm{2}.\:{u}=\mathrm{2}{t}+\sqrt{\mathrm{4}{t}^{\mathrm{2}} −\mathrm{1}} \\ $$ Commented…
Question Number 59732 by maxmathsup by imad last updated on 14/May/19 $${find}\:{I}_{{n}} =\int\:\:\frac{{dx}}{{sin}^{{n}} {x}}\:\:{with}\:\:{n}\:{integr}\:{natural}. \\ $$ Answered by tanmay last updated on 14/May/19 $${I}_{{n}} =\int{cosec}^{{n}}…
Question Number 125234 by mathmax by abdo last updated on 09/Dec/20 $$\mathrm{calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{\mathrm{dx}}{\mathrm{x}+\mathrm{2}+\sqrt{\mathrm{x}^{\mathrm{2}} +\mathrm{x}+\mathrm{1}}} \\ $$ Commented by john_santu last updated on 10/Dec/20 $${using}\:{Second}\:{Euler}\:{substitution}…
Question Number 125233 by mathmax by abdo last updated on 09/Dec/20 $$\mathrm{calculate}\:\mathrm{u}_{\mathrm{nm}} =\int_{\mathrm{0}} ^{\infty} \mathrm{e}^{−\mathrm{nx}} \mathrm{ln}\left(\mathrm{1}+\mathrm{e}^{\mathrm{mx}} \right)\mathrm{dx} \\ $$$$\mathrm{find}\:\sum_{\mathrm{n}\geqslant\mathrm{0}\:\mathrm{and}\:\mathrm{m}\geqslant\mathrm{0}} \:\:\mathrm{u}_{\mathrm{nm}} \\ $$ Terms of Service…