given-function-f-x-f-x-2-for-x-R-if-0-2-f-x-dx-B-what-value-of-3-7-f-x-8-dx-

Question Number 77687 by jagoll last updated on 09/Jan/20

$${given}\:{function} \\ $$$${f}\left({x}\right)\:={f}\left({x}+\mathrm{2}\right)\:{for}\:\forall{x}\in\mathbb{R} \\ $$$${if}\:\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}{f}\left({x}\right){dx}={B}\:,\:{what}\:{value}\:{of} \\ $$$$\underset{\mathrm{3}} {\overset{\mathrm{7}} {\int}}\:{f}\left({x}+\mathrm{8}\right){dx}\:=\:? \\ $$

Commented by mr W last updated on 09/Jan/20

∫_3 ^7 f(x+8)dx =∫_3 ^7 f(x+4×2)dx =∫_3 ^7 f(x)dx =∫_3 ^7 f(x−3)d(x−3) =∫_0 ^4 f(x)dx =∫_0 ^2 f(x)dx+∫_2 ^4 f(x)dx =∫_0 ^2 f(x)dx+∫_2 ^4 f(x−2)d(x−2) =∫_0 ^2 f(x)dx+∫_0 ^2 f(x)dx =2∫_0 ^2 f(x)dx ⇒2B generally if f(x)=f(x+T) and ∫_0 ^T f(x)dx=B, ∫_a ^(a+nT) f(x+mT)dx=n∫_0 ^T f(x)dx=nB with n,m=integer

$$\underset{\mathrm{3}} {\overset{\mathrm{7}} {\int}}\:{f}\left({x}+\mathrm{8}\right){dx} \\ $$$$=\underset{\mathrm{3}} {\overset{\mathrm{7}} {\int}}\:{f}\left({x}+\mathrm{4}×\mathrm{2}\right){dx} \\ $$$$=\underset{\mathrm{3}} {\overset{\mathrm{7}} {\int}}\:{f}\left({x}\right){dx} \\ $$$$=\underset{\mathrm{3}} {\overset{\mathrm{7}} {\int}}\:{f}\left({x}−\mathrm{3}\right){d}\left({x}−\mathrm{3}\right) \\ $$$$=\underset{\mathrm{0}} {\overset{\mathrm{4}} {\int}}\:{f}\left({x}\right){dx} \\ $$$$=\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx}+\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\:{f}\left({x}\right){dx} \\ $$$$=\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx}+\underset{\mathrm{2}} {\overset{\mathrm{4}} {\int}}\:{f}\left({x}−\mathrm{2}\right){d}\left({x}−\mathrm{2}\right) \\ $$$$=\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx}+\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx} \\ $$$$=\mathrm{2}\underset{\mathrm{0}} {\overset{\mathrm{2}} {\int}}\:{f}\left({x}\right){dx} \\ $$$$\Rightarrow\mathrm{2}{B} \\ $$$$ \\ $$$${generally} \\ $$$${if}\:{f}\left({x}\right)={f}\left({x}+{T}\right)\:{and}\:\underset{\mathrm{0}} {\overset{{T}} {\int}}\:{f}\left({x}\right){dx}={B}, \\ $$$$\underset{{a}} {\overset{{a}+{nT}} {\int}}\:{f}\left({x}+{mT}\right){dx}={n}\underset{\mathrm{0}} {\overset{{T}} {\int}}\:{f}\left({x}\right){dx}={nB} \\ $$$${with}\:{n},{m}={integer} \\ $$