Question Number 31838 by Joel578 last updated on 15/Mar/18 $$\mathrm{Given} \\ $$$${f}\left({x}\right)\:=\:\frac{\mathrm{3}}{\mathrm{16}}\:\left(\int_{\mathrm{0}} ^{\mathrm{1}} \:{f}\left({x}\right){dx}\right){x}^{\mathrm{2}} \:−\:\frac{\mathrm{9}}{\mathrm{10}}\left(\int_{\mathrm{0}} ^{\mathrm{2}} \:{f}\left({x}\right){dx}\right){x}\:+\:\mathrm{2}\left(\int_{\mathrm{0}} ^{\mathrm{3}} \:{f}\left({x}\right){dx}\right)\:+\:\mathrm{4} \\ $$$$\mathrm{Solve} \\ $$$$\underset{\mathrm{t}\rightarrow\mathrm{0}} {\mathrm{lim}}\:\frac{\mathrm{2}{t}\:+\:\left(\int_{{f}\left(\mathrm{2}\right)\:+\:\mathrm{2}} ^{{f}^{−\mathrm{1}}…
Question Number 97361 by student work last updated on 07/Jun/20 $$\int\frac{\mathrm{xdx}}{\mathrm{sin}\:^{\mathrm{2}} \mathrm{x}−\mathrm{3}}=? \\ $$ Commented by student work last updated on 07/Jun/20 $$\mathrm{who}\:\mathrm{is}\:\mathrm{intellagent}? \\ $$…
Question Number 97369 by M±th+et+s last updated on 07/Jun/20 $$\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…