Question Number 207652 by universe last updated on 22/May/24 $$\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\left(\mathrm{1}+{x}^{\mathrm{3}} \right){dx}\:\:=\:?{and}\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\:\left(\mathrm{1}+{x}^{\mathrm{4}} \right){dx}\:=\:? \\ $$$$\:\:\mathrm{and}\:\mathrm{if}\:\mathrm{possible}\:\mathrm{then}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:\:\mathrm{p} \\ $$$$\:\mathrm{p}\:\:\:=\:\:\:\int_{\mathrm{0}} ^{\mathrm{1}} \mathrm{log}\left(\mathrm{1}+{x}^{{n}} \right){dx}\:=\:?\:\:\:\:\:\:{n}\in\mathbb{N} \\ $$…
Question Number 207620 by Ghisom last updated on 21/May/24 $$\mathrm{prove}\:\mathrm{that}\:\underset{−{a}} {\overset{{a}} {\int}}\:\frac{{dx}}{{x}^{{n}} +\mathrm{1}+\sqrt{{x}^{\mathrm{2}{n}} +\mathrm{1}}}={a} \\ $$ Answered by Berbere last updated on 21/May/24 $$\int_{−{a}} ^{{a}}…
Question Number 207582 by sniper237 last updated on 19/May/24 $$\int_{\mathrm{0}} ^{\pi} \:{ln}\left({sinx}\right){dx}=−\pi{ln}\mathrm{2} \\ $$$$\int_{\mathrm{0}} ^{\mathrm{1}} {ln}\Gamma\left({x}\right){dx}\:=\:{ln}\left(\mathrm{2}\pi\right) \\ $$ Answered by mathzup last updated on 21/May/24…
Question Number 207565 by mathzup last updated on 18/May/24 $${find}\:\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \:\frac{{x}^{\mathrm{2}} }{{tan}^{\mathrm{2}} {x}}{dx} \\ $$ Answered by sniper237 last updated on 19/May/24 $$\int_{\mathrm{0}} ^{\pi/\mathrm{2}}…
Question Number 207424 by MetaLahor1999 last updated on 14/May/24 $${f}_{{n}} \left({x}\right):=\int{e}^{\frac{\mathrm{2}{x}}{\mathrm{3}}} \frac{{cos}\left({x}\right)}{\:\left({cos}\left({x}\right)+{sin}\left({x}\right)\right)^{\frac{{n}}{\mathrm{3}}} }{dx}=…? \\ $$$${for}\:{n}=\mathrm{1},\:{i}\:{found}\: \\ $$$$\:\:\:\:\:\:{f}_{\mathrm{1}} \left({x}\right)=\frac{\mathrm{3}}{\mathrm{4}}{e}^{\frac{\mathrm{2}{x}}{\mathrm{3}}} \left({cos}\left({x}\right)+{sin}\left({x}\right)\right)^{\frac{\mathrm{2}}{\mathrm{3}}} +\:{C} \\ $$$${is}\:{there}\:{any}\:{ideas}\:{for}\:{a}\:{general}\:{case}\:{or} \\ $$$${the}\:{case}\:{n}=\mathrm{2}? \\…
Question Number 207383 by Shrodinger last updated on 13/May/24 $$\int\frac{{ln}\left({x}^{\mathrm{2}} +{sin}\left({sin}\left({e}^{{x}} \right)\right)\right)}{\:\sqrt{{x}+{tan}\left({ln}\left({x}\right)\right)}}{dx} \\ $$ Commented by Frix last updated on 13/May/24 $$\mathrm{Impossible}. \\ $$ Commented…
Question Number 207382 by efronzo1 last updated on 13/May/24 Answered by sniper237 last updated on 13/May/24 $$\overset{{X}=^{\mathrm{3}} \sqrt{{x}−\mathrm{2}}} {=}\underset{{X}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{{X}^{\mathrm{6}} +\mathrm{2}{X}^{\mathrm{3}} +{X}}{\:^{\mathrm{3}} \sqrt{\mathrm{4}−\mathrm{2}\sqrt{\mathrm{3}{X}^{\mathrm{3}} +\mathrm{4}}−{X}^{\mathrm{3}} \sqrt{\mathrm{3}{X}^{\mathrm{3}}…
Question Number 207352 by NasaSara last updated on 12/May/24 $${calculate}: \\ $$$$\:\int_{\frac{\Pi}{\mathrm{4}}} ^{\frac{\Pi}{\mathrm{2}}} \lfloor{cot}\left({x}\right)\rfloor\:{dx} \\ $$ Commented by NasaSara last updated on 12/May/24 $${thank}\:{you} \\…
Question Number 207359 by Shrodinger last updated on 12/May/24 $$\int\frac{{ln}\left({x}^{\mathrm{2}} +{sin}\left({sin}\left({e}^{{x}} \right)\right)\right)}{\:\sqrt{{x}+{tan}\left({ln}\left({x}\right)\right)}}{dx} \\ $$ Answered by Berbere last updated on 12/May/24 $${it}\:{semms}\:{non}\:{close}\:{forme}\: \\ $$ Commented…
Question Number 207354 by NasaSara last updated on 12/May/24 Commented by mr W last updated on 12/May/24 $${there}\:{are}\:{integrals}\:{like}\:{following} \\ $$$$\int\int…\int\int{f}\left({x}_{\mathrm{1}} ,{x}_{\mathrm{2}} ,…,{x}_{{n}} \right){dx}_{\mathrm{1}} {dx}_{\mathrm{2}} …{dx}_{{n}}…