Question Number 125511 by sandy_delta last updated on 11/Dec/20
$$\mathrm{if}\:: \\ $$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(−\mathrm{x}\right)\:\mathrm{for}\:\mathrm{x}\in\mathbb{R} \\ $$$$\underset{−\mathrm{3}} {\overset{\mathrm{3}} {\int}}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{0} \\ $$$$\underset{−\mathrm{2}} {\overset{\mathrm{3}} {\int}}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{5} \\ $$$$\mathrm{then}\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:? \\ $$
Commented by liberty last updated on 11/Dec/20
$$\Rightarrow\underset{−\mathrm{3}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right){dx}=\mathrm{2}\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right){dx}=\mathrm{0}\:;\:\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right){dx}=\mathrm{0} \\ $$$$\underset{−\mathrm{2}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right){dx}=\underset{−\mathrm{2}} {\overset{\mathrm{0}} {\int}}{f}\left({x}\right){dx}+\underset{\mathrm{0}} {\overset{\mathrm{3}} {\int}}{f}\left({x}\right){dx}\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:=\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}+\mathrm{0} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\mathrm{5}\:\:\:\:\:\:=\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx} \\ $$
Commented by sandy_delta last updated on 12/Dec/20
$$\mathrm{thank}\:\mathrm{you} \\ $$