Question Number 40657 by Raj Singh last updated on 25/Jul/18
Answered by math khazana by abdo last updated on 25/Jul/18
$${let}\:{I}\:=\:\int\:\frac{{a}+{x}}{{a}−{x}}{dx} \\ $$$${I}\:=\:\int\:\frac{{a}−{x}+\mathrm{2}{x}}{{a}−{x}}{dx}\:=\int\:{dx}\:\:+\mathrm{2}\:\int\:\frac{{x}}{{a}−{x}}{dx} \\ $$$$={x}\:−\mathrm{2}\:\int\:\frac{{a}−{x}−{a}}{{a}−{x}}{dx}=\:{x}−\mathrm{2}\:\int\:{dx}\:\:+\mathrm{2}{a}\:\int\:\:\frac{{dx}}{{a}−{x}} \\ $$$$=−{x}\:\:+\mathrm{2}{a}\:{ln}\mid{a}−{x}\mid\:+{c} \\ $$$${I}\:=\mathrm{2}{a}\:{ln}\mid{a}−{x}\mid\:−{x}\:+{c}\:\:. \\ $$