Question Number 114302 by mnjuly1970 last updated on 18/Sep/20 $$\:\:\:\:\:\:\:\:….\:{calculus}\:…. \\ $$$$\:\:\:\:{evaluate}\:::: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$${i}::\int_{\mathrm{0}} ^{\:\mathrm{1}} {t}^{\mathrm{2}} {ln}\left({t}\right){ln}\left(\mathrm{1}−{t}\right){dt}=??? \\ $$$${ii}:::\:\psi^{'} \left(\frac{\mathrm{1}}{\mathrm{4}}\right)=??? \\ $$$${iii}:::\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{8}}}…
Question Number 48757 by sandeepkeshari0797@gmail.com last updated on 28/Nov/18 Commented by maxmathsup by imad last updated on 28/Nov/18 $${method}\:{with}\:{one}\:{parametr}\:{let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\infty} \:\:\frac{{sint}}{{t}}\:{e}^{−{tx}} {dt}\:{with}\:{x}\geqslant\mathrm{0}\:{we}\:{have} \\ $$$${f}^{'} \left({x}\right)=−\int_{\mathrm{0}}…
Question Number 48725 by Tawa1 last updated on 27/Nov/18 $$\int\:\int\:\:\sqrt{\mathrm{x}^{\mathrm{2}} \:+\:\mathrm{y}^{\mathrm{2}} }\:\:\:\mathrm{dx}\:\mathrm{dy},\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sqrt{\mathrm{3y}}\:\:\:\leqslant\:\:\mathrm{x}\:\:\leqslant\:\:\sqrt{\mathrm{4}\:−\:\mathrm{y}^{\mathrm{2}} }\:\:,\:\:\:\:\:\:\:\:\:\mathrm{0}\:\leqslant\:\mathrm{y}\:\leqslant\:\mathrm{2} \\ $$ Commented by Abdo msup. last updated on 27/Nov/18 $${let}\:{I}\:=\int\int_{{D}} \:\:\sqrt{{x}^{\mathrm{2}}…
Question Number 48719 by Abdo msup. last updated on 27/Nov/18 $${find}\:\:\int\:\:\:\:\frac{{x}−\mathrm{2}}{\:\sqrt{{x}^{\mathrm{2}} +\mathrm{4}{x}−\mathrm{3}}}{dx} \\ $$ Commented by Abdo msup. last updated on 27/Nov/18 $${I}=\int\:\:\frac{{x}−\mathrm{2}}{\:\sqrt{{x}^{\mathrm{2}} \:+\mathrm{4}{x}+\mathrm{4}−\mathrm{7}}}\:=\int\:\:\frac{{x}−\mathrm{2}}{\:\sqrt{\left({x}+\mathrm{2}\right)^{\mathrm{2}} −\mathrm{7}}}{dx}…
Question Number 48720 by Abdo msup. last updated on 27/Nov/18 $${calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\:\:\frac{{x}^{\mathrm{2}} \:−\mathrm{2}{cosx}+\mathrm{1}}{{x}^{\mathrm{4}} \:+{x}^{\mathrm{2}} \:+\mathrm{1}}{dx} \\ $$ Commented by Abdo msup. last updated on…
Question Number 48717 by Abdo msup. last updated on 27/Nov/18 $${let}\:\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}+{xtant}\right){dt} \\ $$$$\left.\mathrm{1}\right)\:{find}\:{f}\left({x}\right)\:{at}\:{a}\:{simple}\:{form} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{4}}} {ln}\left(\mathrm{1}+\mathrm{2}{tan}\left({t}\right)\right){dt} \\ $$ Commented by maxmathsup by…
Question Number 48718 by Abdo msup. last updated on 27/Nov/18 $${let}\:{I}_{{n}} =\int_{\mathrm{0}} ^{\mathrm{1}} \left(\mathrm{1}−{t}^{\mathrm{2}} \right)^{{n}} {dt} \\ $$$$\left.\mathrm{1}\right)\:{calculate}\:{I}_{{n}} \:{by}\:{recurrence} \\ $$$$\left.\mathrm{2}\right){find}\:{the}\:{value}\:{of}\:\sum_{{k}=\mathrm{0}} ^{{n}} \:\frac{\left(−\mathrm{1}\right)^{{k}} }{\mathrm{2}{k}+\mathrm{1}}{C}_{{n}} ^{{k}}…
Question Number 48715 by Abdo msup. last updated on 27/Nov/18 $$\left.\mathrm{1}\right)\:{find}\:\:{f}\left(\lambda\right)\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{dx}}{\mathrm{2}+{e}^{−\lambda{x}} }\:\:{with}\:\lambda>\mathrm{0} \\ $$$$\left.\mathrm{2}\right){calculate}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{x}}{\left(\mathrm{2}+{e}^{−\lambda{x}} \right)^{\mathrm{2}} }{dx} \\ $$$$\left.\mathrm{3}\right)\:{find}\:{the}\:{value}\:{of}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\:\:\frac{{dx}}{\mathrm{2}\:+{e}^{−{x}\sqrt{\mathrm{3}}} }{dx}\:{and}\:\int_{\mathrm{0}}…
Question Number 48703 by cesar.marval.larez@gmail.com last updated on 27/Nov/18 Answered by tanmay.chaudhury50@gmail.com last updated on 27/Nov/18 $$\left.\mathrm{3}\right)\int{sin}^{\mathrm{3}} \left(\mathrm{2}{x}\right){cos}\left(\mathrm{2}{x}\right){dx} \\ $$$${t}={sin}\mathrm{2}{x}\:\:\:{dt}=\mathrm{2}{cos}\mathrm{2}{xdx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}\int{t}^{\mathrm{3}} {dt} \\ $$$$=\frac{\mathrm{1}}{\mathrm{2}}×\frac{{t}^{\mathrm{4}}…
Question Number 48667 by maxmathsup by imad last updated on 26/Nov/18 $${find}\:\int_{\mathrm{0}} ^{\mathrm{1}} \:\:\frac{{arctan}\left({x}\right)}{\:\sqrt{\mathrm{1}+{x}^{\mathrm{2}} }}\:{dx}\:. \\ $$ Commented by Abdo msup. last updated on 02/Dec/18…