Question Number 54031 by qw last updated on 28/Jan/19 | ||
$$\mathrm{If}\:\:{f}\left({a}+{b}−{x}\right)=\:{f}\left({x}\right),\:\mathrm{then}\:\underset{{a}} {\overset{{b}} {\int}}\:{x}\:{f}\left({x}\right)\:{dx}\:= \\ $$ | ||
Answered by tanmay.chaudhury50@gmail.com last updated on 28/Jan/19 | ||
$${I}=\int_{{a}} ^{{b}} {xf}\left({x}\right){dx} \\ $$$${I}=\int_{{a}} ^{{b}} \left({a}+{b}−{x}\right){f}\left({a}+{b}−{x}\right){dx} \\ $$$${I}=\int_{{a}} ^{{b}} \left({a}+{b}−{x}\right){f}\left({x}\right){dx}\left[{f}\left({a}+{b}−{x}\right)={f}\left({x}\right)\right] \\ $$$$\mathrm{2}{I}=\int_{{a}} ^{{b}} {xf}\left({x}\right){dx}+\int_{{a}} ^{{b}} \left({a}+{b}−{x}\right){f}\left({x}\right){dx} \\ $$$$\mathrm{2}{I}=\int_{{a}} ^{{b}} \left({a}+{b}−{x}+{x}\right){f}\left({x}\right){dx} \\ $$$${I}=\frac{{a}+{b}}{\mathrm{2}}\int_{{a}} ^{{b}} {f}\left({x}\right){dx} \\ $$$${others}\:{pls}\:{check}... \\ $$ | ||
Commented by maxmathsup by imad last updated on 28/Jan/19 | ||
$${correct}\:{sir}\:{Tanmay}. \\ $$ | ||
Commented by tanmay.chaudhury50@gmail.com last updated on 28/Jan/19 | ||
$${thank}\:{you}\:{sir}... \\ $$ | ||