Question Number 131550 by liberty last updated on 06/Feb/21
![Given f(x)=f(x+4) ∀x∈R If ∫_5 ^7 f(x)dx = T ; ∫_3 ^5 f(x)dx = L then ∫_2 ^(10) f(x) dx =?](https://www.tinkutara.com/question/Q131550.png)
$$\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}+\mathrm{4}\right)\:\forall\mathrm{x}\in\mathbb{R} \\ $$$$\:\mathrm{If}\:\int_{\mathrm{5}} ^{\mathrm{7}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\mathrm{T}\:;\:\int_{\mathrm{3}} ^{\mathrm{5}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\mathrm{L} \\ $$$$\mathrm{then}\:\int_{\mathrm{2}} ^{\mathrm{10}} \mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=? \\ $$
Answered by talminator2856791 last updated on 06/Feb/21
![∫_2 ^( 10) f(x)dx = ∫_2 ^( 3) f(x)dx+∫_3 ^( 5) f(x)dx+∫_5 ^( 7) f(x)dx+∫_7 ^( 10) f(x)dx ∫_2 ^( 3) f(x)dx = ∫_6 ^( 7) f(x)dx ∫_7 ^( 10) f(x)dx = ∫_3 ^( 6) f(x)dx ∫_3 ^( 6) f(x)dx + ∫_6 ^( 7) f(x)dx = ∫_3 ^( 7) f(x)dx ∫_3 ^( 7) f(x)dx = ∫_3 ^( 5) f(x)dx + ∫_5 ^( 7) f(x)dx ∫_3 ^( 7) f(x)dx = T+ L ∫_2 ^( 10) f(x)dx = 2T+2L](https://www.tinkutara.com/question/Q131561.png)
$$\: \\ $$$$\:\int_{\mathrm{2}} ^{\:\mathrm{10}} {f}\left({x}\right){dx}\:=\:\int_{\mathrm{2}} ^{\:\mathrm{3}} {f}\left({x}\right){dx}+\int_{\mathrm{3}} ^{\:\mathrm{5}} {f}\left({x}\right){dx}+\int_{\mathrm{5}} ^{\:\mathrm{7}} {f}\left({x}\right){dx}+\int_{\mathrm{7}} ^{\:\mathrm{10}} {f}\left({x}\right){dx} \\ $$$$\:\int_{\mathrm{2}} ^{\:\mathrm{3}} {f}\left({x}\right){dx}\:=\:\int_{\mathrm{6}} ^{\:\mathrm{7}} {f}\left({x}\right){dx} \\ $$$$\:\int_{\mathrm{7}} ^{\:\mathrm{10}} {f}\left({x}\right){dx}\:=\:\int_{\mathrm{3}} ^{\:\mathrm{6}} {f}\left({x}\right){dx} \\ $$$$\:\int_{\mathrm{3}} ^{\:\mathrm{6}} {f}\left({x}\right){dx}\:+\:\int_{\mathrm{6}} ^{\:\mathrm{7}} {f}\left({x}\right){dx}\:=\:\int_{\mathrm{3}} ^{\:\mathrm{7}} {f}\left({x}\right){dx} \\ $$$$\:\int_{\mathrm{3}} ^{\:\mathrm{7}} {f}\left({x}\right){dx}\:=\:\int_{\mathrm{3}} ^{\:\mathrm{5}} {f}\left({x}\right){dx}\:+\:\int_{\mathrm{5}} ^{\:\mathrm{7}} {f}\left({x}\right){dx} \\ $$$$\:\int_{\mathrm{3}} ^{\:\mathrm{7}} {f}\left({x}\right){dx}\:=\:\mathrm{T}+\:\mathrm{L} \\ $$$$\: \\ $$$$\:\int_{\mathrm{2}} ^{\:\mathrm{10}} {f}\left({x}\right){dx}\:=\:\mathrm{2T}+\mathrm{2L} \\ $$$$\: \\ $$