Question Number 87306 by abdomathmax last updated on 03/Apr/20 $${calculate}\:{by}\:{complex}\:{method}\:\int_{\mathrm{1}} ^{+\infty} \:\frac{{xdx}}{{x}^{\mathrm{4}} \:+\mathrm{1}} \\ $$ Commented by Ar Brandon last updated on 03/Apr/20 $${My}\:\:{suggestion} \\…
Question Number 87301 by M±th+et£s last updated on 03/Apr/20 Commented by mathmax by abdo last updated on 04/Apr/20 $${let}\:\:{I}\:=\int_{\mathrm{1}} ^{+\infty} \:\frac{{dx}}{\mathrm{2}\left[{x}\right]^{\mathrm{2}} \:+\left[{x}\right]}\:\Rightarrow\:{I}\:=\sum_{{n}=\mathrm{1}} ^{\infty} \:\:\int_{{n}} ^{{n}+\mathrm{1}}…
Question Number 152839 by mnjuly1970 last updated on 02/Sep/21 $$ \\ $$$$\:\:\:\mathrm{Prove}\:\:\mathrm{that}\:: \\ $$$$ \\ $$$$\:\Omega=\int_{\mathrm{0}} ^{\:\mathrm{1}} \frac{\:\mathrm{ln}^{\:\mathrm{3}} \:\left(\mathrm{1}\:+\:{x}\:\right)}{{x}^{\:\mathrm{2}} }{dx}\:=\:\frac{\mathrm{3}}{\mathrm{4}}\:\zeta\:\left(\mathrm{3}\:\right)−\:\mathrm{2ln}^{\:\mathrm{3}} \left(\:\mathrm{2}\:\right)\:\:\:\:\:\:\:\:\blacksquare \\ $$$$\:\:\:\:\:\:\:\mathrm{Prepared}\:\mathrm{by}:\:\:\:\:\:\:\mathrm{M}.\mathrm{N} \\ $$$$…
Question Number 87296 by ajfour last updated on 03/Apr/20 $${If}\:\:{ellipse}\:\:\frac{{x}^{\mathrm{2}} }{{a}^{\mathrm{2}} }+\frac{{y}^{\mathrm{2}} }{{b}^{\mathrm{2}} }=\mathrm{1}\:\:\left({a}>{b}\right) \\ $$$${is}\:{rotated}\:{about}\:{x}-{axis},\:{find}\:{the} \\ $$$${surface}\:{of}\:{the}\:{solid}\:{of}\:{revolution}. \\ $$ Answered by mr W last…
Question Number 87298 by ajfour last updated on 03/Apr/20 $${If}\:{y}=\mathrm{sin}\:{x}\:,\:\:{x}=\mathrm{0}\:{to}\:{x}=\mathrm{2}\pi\:{is} \\ $$$${revolved}\:{about}\:{the}\:{x}-{axis},\:{find} \\ $$$${the}\:{surface}\:{of}\:{the}\:{solid}\:{of} \\ $$$${revolution}. \\ $$ Answered by ajfour last updated on 04/Apr/20…
Question Number 87279 by Ar Brandon last updated on 03/Apr/20 $$\int\frac{{x}^{\mathrm{2}} }{\mathrm{1}+{x}^{\mathrm{4}} }{dx} \\ $$ Commented by abdomathmax last updated on 03/Apr/20 $${complex}\:{method}\:\:{let}\:{decompose}\:{F}\left({x}\right)=\frac{{x}^{\mathrm{2}} }{{x}^{\mathrm{4}} \:+\mathrm{1}}…
Question Number 21721 by Isse last updated on 02/Oct/17 $$\int_{\mathrm{0}} ^{\pi/\mathrm{2}} {sin}^{\mathrm{2}} {xcos}^{\mathrm{3}} {xdx} \\ $$ Answered by sma3l2996 last updated on 02/Oct/17 $$=\int_{\mathrm{0}} ^{\pi/\mathrm{2}}…
Question Number 21720 by Isse last updated on 01/Oct/17 $$\int{sin}^{\mathrm{5}} \theta{d}\theta \\ $$ Answered by sma3l2996 last updated on 02/Oct/17 $$=\int\left(\mathrm{1}−{cos}^{\mathrm{2}} {x}\right)^{\mathrm{2}} {sinxdx}=\int\left({sinx}−\mathrm{2}{sinxcos}^{\mathrm{2}} {x}+{sinxcos}^{\mathrm{4}} {x}\right){dx}…
Question Number 21708 by Isse last updated on 01/Oct/17 $$\int_{\mathrm{0}} ^{\mathrm{0}.\mathrm{5}} \mathrm{2}{tan}^{\mathrm{2}} \mathrm{2}{tdt} \\ $$ Answered by $@ty@m last updated on 01/Oct/17 $$\int_{\mathrm{0}} ^{\mathrm{0}.\mathrm{5}} \mathrm{2}\left(\mathrm{sec}\:^{\mathrm{2}}…
Question Number 21707 by Isse last updated on 01/Oct/17 $$\int_{\pi/\mathrm{6}} ^{\pi/\mathrm{3}} \frac{\mathrm{1}}{\mathrm{2}}{cot}^{\mathrm{2}} \mathrm{2}\theta{d}\theta \\ $$ Commented by Tikufly last updated on 01/Oct/17 $$\mathrm{I}=\frac{\mathrm{1}}{\mathrm{2}}\int_{\pi/\mathrm{6}} ^{\pi/\mathrm{3}} \left(\mathrm{cosec}^{\mathrm{2}}…