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Question Number 131146 by EDWIN88 last updated on 02/Feb/21

Given f(x)=f(x+6) ∀x∈R  If ∫_(−2) ^( 2) f(x)dx=−2 and ∫_(−2) ^( 4) f(x)dx=2  find the value of ∫_4 ^( 10) f(x)dx.

$${Given}\:{f}\left({x}\right)={f}\left({x}+\mathrm{6}\right)\:\forall{x}\in\mathbb{R} \\ $$$${If}\:\int_{−\mathrm{2}} ^{\:\mathrm{2}} {f}\left({x}\right){dx}=−\mathrm{2}\:{and}\:\int_{−\mathrm{2}} ^{\:\mathrm{4}} {f}\left({x}\right){dx}=\mathrm{2} \\ $$$${find}\:{the}\:{value}\:{of}\:\int_{\mathrm{4}} ^{\:\mathrm{10}} {f}\left({x}\right){dx}. \\ $$

Commented by EDWIN88 last updated on 02/Feb/21

⇒∫_(−2) ^( 4) f(x)dx=∫_(−2) ^( 4) f(x+6)dx=2  ∫_4 ^( 10) f(x)dx = 2

$$\Rightarrow\int_{−\mathrm{2}} ^{\:\mathrm{4}} {f}\left({x}\right){dx}=\int_{−\mathrm{2}} ^{\:\mathrm{4}} {f}\left({x}+\mathrm{6}\right){dx}=\mathrm{2} \\ $$$$\int_{\mathrm{4}} ^{\:\mathrm{10}} {f}\left({x}\right){dx}\:=\:\mathrm{2} \\ $$

Answered by Ar Brandon last updated on 02/Feb/21

Let x−6=u  ⇒I=∫_(−2) ^4 f(u+6)du=∫_(−2) ^4 f(u)du=2

$$\mathrm{Let}\:\mathrm{x}−\mathrm{6}=\mathrm{u} \\ $$$$\Rightarrow\mathcal{I}=\int_{−\mathrm{2}} ^{\mathrm{4}} \mathrm{f}\left(\mathrm{u}+\mathrm{6}\right)\mathrm{du}=\int_{−\mathrm{2}} ^{\mathrm{4}} \mathrm{f}\left(\mathrm{u}\right)\mathrm{du}=\mathrm{2} \\ $$

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