Question Number 159172 by aliibrahim1 last updated on 13/Nov/21
Answered by ajfour last updated on 13/Nov/21
$${Apply}\:{Integration}\:{by}\:{parts};{also} \\ $$$$\int{e}^{{x}} \left\{{f}\left({x}\right)+{f}\:'\left({x}\right)\right\}{dx}={e}^{{x}} {f}\left({x}\right)+{c} \\ $$$${Here}\:\:\:\frac{{y}}{\left({y}+\mathrm{1}\right)^{\mathrm{2}} }=\frac{{y}+\mathrm{1}−\mathrm{1}}{\left({y}+\mathrm{1}\right)^{\mathrm{2}} } \\ $$$$\:\:\:=\:\:\frac{\mathrm{1}}{{y}+\mathrm{1}}+\frac{−\mathrm{1}}{\mathrm{1}+{y}^{\mathrm{2}} }={f}\left({y}\right)+{f}\:'\left({y}\right) \\ $$$${hence}\:\:\int\frac{{e}^{{y}} {ydy}}{\left({y}+\mathrm{1}\right)^{\mathrm{2}} }=\frac{{e}^{{y}} }{{y}+\mathrm{1}}+{c} \\ $$
Commented by aliibrahim1 last updated on 13/Nov/21
$${thx}\:{sir} \\ $$