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

$$givenfunction$$$$f\left(x\right)=f(x+2)for\mathrm{\forall }x\in \mathbb{R}$$$$if\underset{0}{\stackrel{2}{\int }}f\left(x\right)dx=B,whatvalueof$$$$\underset{3}{\stackrel{7}{\int }}f(x+8)dx=?$$

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

$$\underset{3}{\stackrel{7}{\int }}f(x+8)dx$$$$=\underset{3}{\stackrel{7}{\int }}f(x+4\times 2)dx$$$$=\underset{3}{\stackrel{7}{\int }}f\left(x\right)dx$$$$=\underset{3}{\stackrel{7}{\int }}f(x-3)d(x-3)$$$$=\underset{0}{\stackrel{4}{\int }}f\left(x\right)dx$$$$=\underset{0}{\stackrel{2}{\int }}f\left(x\right)dx+\underset{2}{\stackrel{4}{\int }}f\left(x\right)dx$$$$=\underset{0}{\stackrel{2}{\int }}f\left(x\right)dx+\underset{2}{\stackrel{4}{\int }}f(x-2)d(x-2)$$$$=\underset{0}{\stackrel{2}{\int }}f\left(x\right)dx+\underset{0}{\stackrel{2}{\int }}f\left(x\right)dx$$$$=2\underset{0}{\stackrel{2}{\int }}f\left(x\right)dx$$$$\Rightarrow 2B$$$$$$$$generally$$$$iff\left(x\right)=f(x+T)and\underset{0}{\stackrel{T}{\int }}f\left(x\right)dx=B,$$$$\underset{a}{\stackrel{a+nT}{\int }}f(x+mT)dx=n\underset{0}{\stackrel{T}{\int }}f\left(x\right)dx=nB$$$$withn,m=integer$$

Commented by jagoll last updated on 09/Jan/20

$$thanksyousir$$

Commented by jagoll last updated on 09/Jan/20

$$sirWhowaboutf\left(x\right)=f(x+7)\mathrm{\forall }x\in \mathbb{R}$$$$if\underset{2}{\stackrel{4}{\int }}f\left(x\right)dx=5and\underset{3}{\stackrel{4}{\int }}f\left(x\right)dx=2$$$$whatis\underset{9}{\stackrel{10}{\int }}f\left(x\right)dx=?$$

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

$$\underset{9}{\stackrel{10}{\int }}f\left(x\right)dx$$$$=\underset{9}{\stackrel{10}{\int }}f(x-7)d(x-7)$$$$=\underset{2}{\stackrel{3}{\int }}f\left(x\right)dx$$$$=\underset{2}{\stackrel{4}{\int }}f\left(x\right)dx-\underset{3}{\stackrel{4}{\int }}f\left(x\right)dx$$$$=5-2$$$$=3$$

Commented by jagoll last updated on 09/Jan/20

$$thanksyousir$$

Commented by jagoll last updated on 09/Jan/20

$$sirline8is\underset{0}{\stackrel{2}{\int }}f\left(x\right)dx+\underset{0}{\stackrel{2}{\int }}f\left(x\right)dx.iscorrectsir?$$

Commented by jagoll last updated on 09/Jan/20

$$ithinkityourtyposir$$

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

$$yes,nowfixed!$$

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