Question Number 97369 by M±th+et+s last updated on 07/Jun/20 $$\int_{\mathrm{0}} ^{\mathrm{1}} \frac{\left({ln}\left({x}\right)\right)^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{2}} }{dx}=\frac{\pi^{\mathrm{3}} }{\mathrm{16}} \\ $$ Answered by maths mind last updated on 07/Jun/20…
Question Number 162893 by mnjuly1970 last updated on 02/Jan/22 $$ \\ $$$$\:\:\:\:\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \left(\frac{\:{x}^{\:} }{\mathrm{ln}^{\:} \left(\:\mathrm{1}−{x}\:\right)}\right)^{\:\mathrm{2}} {dx}\overset{?} {=}\:\mathrm{ln}\:\left(\frac{\:\mathrm{27}}{\mathrm{16}}\:\right) \\ $$$$\:\:\:\:\:\:\:\:−−−− \\ $$$$ \\ $$ Answered…
Question Number 162894 by mathacek last updated on 02/Jan/22 Answered by Ar Brandon last updated on 02/Jan/22 $$\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{sin}{x}\right){dx}−\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}} \mathrm{ln}\left(\mathrm{cos}{x}\right){dx} \\ $$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{2}}}…
Question Number 162864 by mathacek last updated on 01/Jan/22 Answered by Ar Brandon last updated on 01/Jan/22 $${I}=\int_{\mathrm{0}} ^{\pi} {x}\mathrm{ln}\left(\mathrm{sin}{x}\right){dx}=\int_{\mathrm{0}} ^{\pi} \left(\pi−{x}\right)\mathrm{ln}\left(\mathrm{sin}{x}\right){dx} \\ $$$$\:\:\:=\pi\int_{\mathrm{0}} ^{\pi}…
Question Number 31787 by neel1974 last updated on 14/Mar/18 $$\int\frac{\mathrm{4}{x}−\mathrm{3}}{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{8}}{dx} \\ $$ Answered by sma3l2996 last updated on 14/Mar/18 $${A}=\int\frac{\mathrm{4}{x}−\mathrm{3}}{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{8}}{dx}=\mathrm{2}\int\frac{\mathrm{2}{x}−\mathrm{3}/\mathrm{2}}{{x}^{\mathrm{2}} +\mathrm{3}{x}+\mathrm{8}}{dx} \\ $$$$=\mathrm{2}\int\left(\frac{\mathrm{2}{x}+\mathrm{3}}{{x}^{\mathrm{2}}…
Question Number 97322 by student work last updated on 07/Jun/20 $$\int\mathrm{sin}\:^{\mathrm{4}} \mathrm{x}\centerdot\mathrm{cos}\:^{\mathrm{5}} \mathrm{xdx}=? \\ $$ Answered by bemath last updated on 08/Jun/20 $$\int\:\mathrm{sin}\:^{\mathrm{4}} {x}\:\left(\mathrm{1}−\mathrm{sin}\:^{\mathrm{2}} {x}\right)^{\mathrm{2}}…
Question Number 31747 by abdo imad last updated on 13/Mar/18 $${let}\:{give}\:\mid\lambda\mid<\mathrm{1}\:{and}\:{u}_{{n}} =\:\int_{\mathrm{0}} ^{\pi} \:\:\frac{{cos}\left({nx}\right)}{\mathrm{1}−\mathrm{2}\lambda\:{cosx}\:+\lambda^{\mathrm{2}} } \\ $$$${prove}\:{that}\:\sum_{{n}=\mathrm{0}} ^{\infty} \:{u}_{{n}} \:{is}\:{convergent}\:{and}\:{find}\:{its}\:{sum}\:. \\ $$ Commented by abdo…
Question Number 162811 by mnjuly1970 last updated on 01/Jan/22 $$ \\ $$$$\:\:\:\:\:\mathrm{I}\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:{tan}^{\:−\mathrm{1}} \:\left({x}\:\right)}{\left(\:\mathrm{1}+{x}^{\:\mathrm{2}} \:\right)^{\:\mathrm{2}} }\:{dx}\:=\:? \\ $$$$\:\:\:\:\:\:−−−−−−−−−− \\ $$ Answered by amin96 last…
Question Number 97239 by john santu last updated on 07/Jun/20 $$\int\:\frac{\mathrm{sec}\:^{\mathrm{3}} {x}\:{dx}}{\:\sqrt{\mathrm{tan}\:{x}}}\:?\: \\ $$ Commented by MJS last updated on 07/Jun/20 $${t}=\sqrt{\mathrm{tan}\:{x}}\:\mathrm{leads}\:\mathrm{to}\:\mathrm{2}\int\sqrt{{t}^{\mathrm{4}} +\mathrm{1}}{dt}\:\mathrm{and}\:\mathrm{we}\:\mathrm{cannot} \\ $$$$\mathrm{solve}\:\mathrm{this}\:\mathrm{using}\:\mathrm{elementary}\:\mathrm{calculus}…
Question Number 97235 by bobhans last updated on 07/Jun/20 $$\underset{\mathrm{0}} {\overset{\mathrm{1}} {\int}}\:\mathrm{ln}\left(\mathrm{x}\right)\:\mathrm{ln}\left(\mathrm{1}−\mathrm{x}\right)\:\mathrm{dx}\:?\: \\ $$ Answered by abdomathmax last updated on 07/Jun/20 $$\mathrm{I}\:=\int_{\mathrm{0}} ^{\mathrm{1}} \:\mathrm{lnxln}\left(\mathrm{1}−\mathrm{x}\right)\mathrm{dx}\:\:\mathrm{we}\:\mathrm{have}\:\mathrm{for}\:\mid\mathrm{x}\mid<\mathrm{1} \\…