Question Number 122919 by bemath last updated on 20/Nov/20
Answered by Dwaipayan Shikari last updated on 20/Nov/20
$$\int_{\mathrm{0}} ^{\frac{\mathrm{1}}{\mathrm{2}}} \frac{{xe}^{\mathrm{2}{x}} }{\left(\mathrm{1}+\mathrm{2}{x}\right)^{\mathrm{2}} }{dx} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{ue}^{{u}} }{\left(\mathrm{1}+{u}\right)^{\mathrm{2}} }{du} \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\int_{\mathrm{0}} ^{\mathrm{1}} \frac{{e}^{{u}} }{\left({u}+\mathrm{1}\right)}−\frac{{e}^{{u}} }{\left({u}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$=\frac{\mathrm{1}}{\mathrm{4}}\left[\frac{{e}^{{u}} }{{u}+\mathrm{1}}\right]_{\mathrm{0}} ^{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{{e}}{\mathrm{2}}−\mathrm{1}\right)=\frac{\mathrm{1}}{\mathrm{8}}\left({e}−\mathrm{2}\right) \\ $$
Answered by liberty last updated on 20/Nov/20
$${let}\:\mathrm{1}+\mathrm{2}{x}\:=\:{u}\:\Rightarrow{x}=\frac{{u}−\mathrm{1}}{\mathrm{2}}\:\wedge\:{dx}\:=\:\frac{\mathrm{1}}{\mathrm{2}}{du} \\ $$$$\phi\:=\:\int\:\frac{\left(\frac{{u}−\mathrm{1}}{\mathrm{2}}\right){e}^{{u}−\mathrm{1}} }{{u}^{\mathrm{2}} }\:\frac{\mathrm{1}}{\mathrm{2}}{du}\:=\:\frac{\mathrm{1}}{\mathrm{4}}\int\:\frac{\left({u}−\mathrm{1}\right){e}^{{u}−\mathrm{1}} }{{u}^{\mathrm{2}} }{du} \\ $$$$\phi\:=\:\frac{\mathrm{1}}{\mathrm{4}{e}}\:\int\:\frac{\left({u}−\mathrm{1}\right){e}^{{u}} }{{u}^{\mathrm{2}} }\:{du}\:=\:\frac{\mathrm{1}}{\mathrm{4}{e}}\left[\int\:\frac{{u}.{e}^{{u}} −{e}^{{u}} }{{u}^{\mathrm{2}} }\:{du}\right] \\ $$$$\phi\:=\:\frac{\mathrm{1}}{\mathrm{4}{e}}\int\:{d}\left(\frac{{e}^{{u}} }{{u}}\right)\:=\:\frac{\mathrm{1}}{\mathrm{4}{e}}\left(\frac{{e}^{{u}} }{{u}}\right)=\frac{{e}^{{u}−\mathrm{1}} }{\mathrm{4}{u}} \\ $$$${put}\:{border}\:\begin{cases}{{u}=\mathrm{2}}\\{{u}=\mathrm{1}}\end{cases} \\ $$$${give}\:\phi\:=\:\frac{\mathrm{1}}{\mathrm{4}}\:\left[\frac{{e}^{{u}−\mathrm{1}} }{{u}}\:\right]_{\mathrm{1}} ^{\mathrm{2}} =\:\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{{e}}{\mathrm{2}}−\mathrm{1}\right).\: \\ $$