Question Number 131550 by liberty last updated on 06/Feb/21
$$\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}+\mathrm{4}\right)\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\:\mathrm{If}\:\int_{\mathrm{5}} ^{\mathrm{7}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\mathrm{T}\:;\:\int_{\mathrm{3}} ^{\mathrm{5}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\mathrm{L} \\ $$$$\mathrm{then}\:\int_{\mathrm{2}} ^{\mathrm{10}} \mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=? \\ $$
Answered by talminator2856791 last updated on 06/Feb/21
$$\: \\ $$$$\:\int_{\mathrm{2}} ^{\:\mathrm{10}} {f}\left({x}\right){dx}\:=\:\int_{\mathrm{2}} ^{\:\mathrm{3}} {f}\left({x}\right){dx}+\int_{\mathrm{3}} ^{\:\mathrm{5}} {f}\left({x}\right){dx}+\int_{\mathrm{5}} ^{\:\mathrm{7}} {f}\left({x}\right){dx}+\int_{\mathrm{7}} ^{\:\mathrm{10}} {f}\left({x}\right){dx} \\ $$$$\:\int_{\mathrm{2}} ^{\:\mathrm{3}} {f}\left({x}\right){dx}\:=\:\int_{\mathrm{6}} ^{\:\mathrm{7}} {f}\left({x}\right){dx} \\ $$$$\:\int_{\mathrm{7}} ^{\:\mathrm{10}} {f}\left({x}\right){dx}\:=\:\int_{\mathrm{3}} ^{\:\mathrm{6}} {f}\left({x}\right){dx} \\ $$$$\:\int_{\mathrm{3}} ^{\:\mathrm{6}} {f}\left({x}\right){dx}\:+\:\int_{\mathrm{6}} ^{\:\mathrm{7}} {f}\left({x}\right){dx}\:=\:\int_{\mathrm{3}} ^{\:\mathrm{7}} {f}\left({x}\right){dx} \\ $$$$\:\int_{\mathrm{3}} ^{\:\mathrm{7}} {f}\left({x}\right){dx}\:=\:\int_{\mathrm{3}} ^{\:\mathrm{5}} {f}\left({x}\right){dx}\:+\:\int_{\mathrm{5}} ^{\:\mathrm{7}} {f}\left({x}\right){dx} \\ $$$$\:\int_{\mathrm{3}} ^{\:\mathrm{7}} {f}\left({x}\right){dx}\:=\:\mathrm{T}+\:\mathrm{L} \\ $$$$\: \\ $$$$\:\int_{\mathrm{2}} ^{\:\mathrm{10}} {f}\left({x}\right){dx}\:=\:\mathrm{2T}+\mathrm{2L} \\ $$$$\: \\ $$