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Question Number 79419 by jagoll last updated on 25/Jan/20

$$\mathrm{given}$$$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}(\mathrm{x}+5)\mathrm{\forall }\mathrm{x}\in \mathbb{R}$$$$\mathrm{If}\int {}_7^9\mathrm{f}\left(\mathrm{x}\right)=\mathrm{t}\mathrm{and}\int {}_2^6\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=$$$${\mathrm{t}}^2+4\mathrm{t}-3.\mathrm{find}\mathrm{the}\mathrm{value}\mathrm{of}\mathrm{t}.$$

Commented by jagoll last updated on 25/Jan/20

$$\underset{7}{\stackrel{9}{\int }}\mathrm{f}(\mathrm{x}-5)\mathrm{d}(\mathrm{x}-5)=\mathrm{t}$$$$\underset{2}{\stackrel{4}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{t}\Rightarrow \underset{2}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=$$$$\underset{2}{\stackrel{4}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+\underset{4}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=2\mathrm{t}$$$$\mathrm{therefore}2\mathrm{t}={\mathrm{t}}^2+4\mathrm{t}-3$$$${\mathrm{t}}^2+2\mathrm{t}-3=0\{\begin{array}{l}\mathrm{t}=-3\\ \mathrm{t}=1\end{array}$$

Commented by jagoll last updated on 25/Jan/20

$$\mathrm{Mr}\mathrm{W},\mathrm{that}\mathrm{right}?$$

Commented by john santu last updated on 25/Jan/20

$$ithinkwrong.$$$$\underset{2}{\stackrel{6}{\int }}f\left(x\right)dx=\underset{2}{\stackrel{6}{\int }}f(x+5)d(x+5)=$$$$\underset{7}{\stackrel{11}{\int }}f\left(x\right)dx=??$$

Commented by mr W last updated on 25/Jan/20

$${it}^{\prime }swrongsir!$$$$\underset{2}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\underset{2}{\stackrel{4}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}+\underset{4}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$$$$=t+\underset{4}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\ne 2\mathrm{t}$$$$since\underset{4}{\stackrel{6}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\ne \underset{2}{\stackrel{4}{\int }}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$$

Commented by mr W last updated on 25/Jan/20

$$again,ifTisperiod,$$$${\int }_a^{a+nT}f\left(x\right)dx={\int }_c^{c+nT}f\left(x\right)dxforanyaandc$$$$but{\int }_a^{a+S}f\left(x\right)dx\ne {\int }_c^{c+S}f\left(x\right)dxifS\ne nTandc\ne a+kT$$

Commented by jagoll last updated on 25/Jan/20

$$\mathrm{thanks}\mathrm{you}\mathrm{mister}$$

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