Question Number 174727 by princeDera last updated on 09/Aug/22 $$\int_{\mathrm{0}} ^{\infty} \frac{{e}^{−{x}^{\mathrm{2}} } }{\left({x}^{\mathrm{2}} +{a}^{\mathrm{2}} \right)^{\mathrm{2}} }{dx} \\ $$ Answered by aleks041103 last updated on…
Question Number 174696 by mnjuly1970 last updated on 08/Aug/22 $$ \\ $$$$\:\:\:\:\:\boldsymbol{{prove}}\:\:\boldsymbol{{that}}\:: \\ $$$$ \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{\Omega}\:=\:\int_{\mathrm{0}} ^{\:\infty} \left(\:\frac{\:\boldsymbol{{x}}}{\:\boldsymbol{{sinh}}\:\left(\boldsymbol{{x}}\right)}\:\right)^{\:\mathrm{3}} \boldsymbol{{dx}}\:=\frac{\boldsymbol{\pi}^{\:\mathrm{2}} }{\mathrm{16}}\:\left(\mathrm{12}−\:\boldsymbol{\pi}^{\:\mathrm{2}} \right)\:\:\:\:\: \\ $$$$\:\:\:\:\:\:\:\:\boldsymbol{{written}}\:\:\boldsymbol{{and}}\:\boldsymbol{{prepared}}\:\boldsymbol{{by}}\::\:\:\boldsymbol{{m}}.\boldsymbol{{n}}\:\:\:\:\:\:\:\: \\ $$$$…
Question Number 43623 by math khazana by abdo last updated on 12/Sep/18 $${let}\:{f}\left({x}\right)=\int_{\mathrm{0}} ^{{x}} \:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{4}} } \\ $$$$\left.\mathrm{1}\right)\:{find}\:\:{a}\:{explicit}\:{form}\:{of}\:{f}\left({x}\right) \\ $$$$\left.\mathrm{2}\right)\:{calculate}\:\int_{\mathrm{0}} ^{\infty} \:\:\frac{{dt}}{\mathrm{1}+{t}^{\mathrm{4}} } \\ $$…
Question Number 174685 by mnjuly1970 last updated on 08/Aug/22 $$ \\ $$$$\:\:\:\:\:{prove}\:{that}\:: \\ $$$$\: \\ $$$$\:\:\:\:\:\Omega\:=\:\int_{\mathrm{0}} ^{\:\infty} \frac{\:\:{x}^{\:\mathrm{2}} }{{cosh}\left({x}\:\right)}\:{dx}\:=\:\frac{\pi^{\:\mathrm{3}} }{\:\mathrm{8}} \\ $$$$ \\ $$ Answered…
Question Number 109136 by EmericGent last updated on 21/Aug/20 $$\int_{\mathrm{0}} ^{\mathrm{1}/\mathrm{2}} \frac{{ln}\left(\mathrm{1}-{t}\right){ln}\left({t}\right)}{{t}}\:{dt} \\ $$$${I}'{m}\:{about}\:{to}\:{give}\:{up} \\ $$ Answered by Sarah85 last updated on 21/Aug/20 $$\int\frac{\mathrm{ln}\:\left(\mathrm{1}−{t}\right)\:\mathrm{ln}\:{t}}{{t}}{dt} \\…
Question Number 109129 by bemath last updated on 21/Aug/20 $$\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{2}} {\int}}\:\frac{\mathrm{ln}\:\left(\mathrm{cos}\:{x}\right)\mathrm{ln}\:\left(\mathrm{sin}\:{x}\right)}{\mathrm{tan}\:{x}}\:{dx} \\ $$ Answered by EmericGent last updated on 21/Aug/20 $$=\:{I}\:=\frac{\mathrm{1}}{\mathrm{2}}\int_{\mathrm{0}} ^{\pi/\mathrm{2}} \frac{{ln}\left(\mathrm{1}-{sin}^{\mathrm{2}} {x}\right){ln}\left({sin}\:{x}\right)}{{sin}\:{x}}\:{cos}\:{x}\:{dx}…
Question Number 43589 by Tawa1 last updated on 12/Sep/18 Commented by maxmathsup by imad last updated on 12/Sep/18 $${let}\:{I}\:=\:\int_{\mathrm{0}} ^{\mathrm{2}\pi} \:\:\:\frac{{cos}\beta}{\mathrm{10}\:+\mathrm{8}{sin}\beta}\:{d}\beta\:\:\:{changement}\:\:{e}^{{i}\beta} ={z}\:{give} \\ $$$${I}\:\:=\:\int_{\mid{z}\mid=\mathrm{1}} \:\:\:\:\:\frac{\frac{{z}+{z}^{−\mathrm{1}}…
Question Number 109101 by bemath last updated on 21/Aug/20 $$\:{Given}\:{a}\:{function}\:{f}\left({x}+\mathrm{3}\right)={f}\left({x}\right) \\ $$$${for}\:\forall{x}\in\mathbb{R}.\:{If}\:\underset{−\mathrm{3}} {\overset{\mathrm{6}} {\int}}{f}\left({x}\right){dx}\:=\:−\mathrm{6}\: \\ $$$${then}\:\underset{\mathrm{3}} {\overset{\mathrm{9}} {\int}}{f}\left({x}\right)\:{dx}\:=\:? \\ $$ Answered by bemath last updated…
Question Number 174628 by cortano1 last updated on 06/Aug/22 $$\:\:\:\Omega\:=\:\int\:\frac{{x}}{\mathrm{1}+\mathrm{csc}\:{x}}\:{dx}\:=? \\ $$ Commented by infinityaction last updated on 06/Aug/22 $$\int\frac{{x}}{\mathrm{1}+\mathrm{csc}{x}}{dx}=\int\frac{{xs}\mathrm{in}{x}}{\mathrm{1}+\mathrm{sin}{x}}{dx} \\ $$$$=\int{x}\frac{\mathrm{1}+\mathrm{sin}{x}−\mathrm{1}}{\mathrm{1}+\mathrm{sin}{x}}{dx}=\int\left({x}−\frac{{x}}{\mathrm{1}+\mathrm{sin}{x}}\right){dx} \\ $$$$\Omega\:=\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}}−\int\frac{{x}}{\underset{{I}}…
Question Number 109097 by bobhans last updated on 21/Aug/20 $$\:\:\frac{\boldsymbol{\flat{o}\flat{hans}}}{\sim\sim\sim\sim\sim} \\ $$$$\underset{\mathrm{1}} {\overset{\mathrm{2}} {\int}}{x}\:\mathrm{sec}^{−\mathrm{1}} \left({x}\right){dx}=? \\ $$ Answered by john santu last updated on 21/Aug/20…