Question Number 57136 by mustakim420 last updated on 30/Mar/19
$$\underset{{a}} {\overset{{b}} {\int}}\:\:\frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left({a}+{b}−{x}\right)}\:{dx}\:= \\ $$
Answered by tanmay.chaudhury50@gmail.com last updated on 30/Mar/19
$${I}=\int_{{a}} ^{{b}} \frac{{f}\left({x}\right)}{{f}\left({x}\right)+{f}\left({a}+{b}−{x}\right)}{dx} \\ $$$${I}=\int_{{a}} ^{{b}} \frac{{f}\left({a}+{b}−{x}\right)}{{f}\left({a}+{b}−{x}\right)+{f}\left({x}\right)}{dx} \\ $$$$\mathrm{2}{I}=\int_{{a}} ^{{b}} \frac{{f}\left({x}\right)+{f}\left({a}+{b}−{x}\right)}{{f}\left({x}\right)+{f}\left({a}+{b}−{x}\right)}{dx} \\ $$$$\mathrm{2}{I}=\int_{{a}} ^{{b}} {dx} \\ $$$$\mathrm{2}{I}={b}−{a} \\ $$$${I}=\frac{{b}−{a}}{\mathrm{2}} \\ $$$${note}\:{formula} \\ $$$$\int_{{a}} ^{{b}} {f}\left({x}\right){dx}=\int_{{a}} ^{{b}} {f}\left({a}+{b}−{x}\right){dx} \\ $$