# Given-f-x-f-x-pi-6-x-R-if-0-pi-6-f-x-dx-T-find-the-value-of-pi-4pi-3-f-x-pi-dx-nice-integral-

Question Number 131496 by benjo_mathlover last updated on 05/Feb/21
$$\mathrm{Given}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}+\frac{\pi}{\mathrm{6}}\right)\:\forall\mathrm{x}\in\mathbb{R} \\$$$$\mathrm{if}\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{6}}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\:\mathrm{T}\:\mathrm{find}\:\mathrm{the}\:\mathrm{value} \\$$$$\mathrm{of}\:\int_{\pi} ^{\:\frac{\mathrm{4}\pi}{\mathrm{3}}} \mathrm{f}\left(\mathrm{x}+\pi\right)\mathrm{dx}. \\$$$$\mathrm{nice}\:\mathrm{integral} \\$$
Answered by talminator2856791 last updated on 05/Feb/21
$$\:\mathrm{2T} \\$$
Commented by benjo_mathlover last updated on 05/Feb/21
$$\mathrm{why}? \\$$
Answered by EDWIN88 last updated on 05/Feb/21
$$\:\Leftrightarrow\:\int_{\pi} ^{\frac{\mathrm{4}}{\mathrm{3}}\pi} \mathrm{f}\left(\mathrm{x}+\pi\right)\mathrm{dx}\:=\:\int_{\pi−\pi} ^{\frac{\mathrm{4}\pi}{\mathrm{3}}−\pi} \mathrm{f}\left(\mathrm{x}+\pi−\pi\right)\:\mathrm{dx}= \\$$$$\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{3}}} \mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\int_{\mathrm{0}} ^{\mathrm{0}+\mathrm{2}\left(\frac{\pi}{\mathrm{6}}\right)} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx} \\$$$$\:=\:\mathrm{2T}\: \\$$$$\left(\mathrm{by}\:\mathrm{theorem}\:\mathrm{If}\:\int_{\mathrm{0}} ^{\:\mathrm{P}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:=\:\mathrm{G}\:\mathrm{then}\:\int_{\mathrm{0}} ^{\:\mathrm{n}.\mathrm{P}} \mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{n}.\mathrm{G}\:\right) \\$$$$\mathrm{for}\:\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}\left(\mathrm{x}+\mathrm{P}\right)\:\forall\mathrm{x}\in\mathbb{R} \\$$
Commented by benjo_mathlover last updated on 05/Feb/21
$$\mathrm{thank}\:\mathrm{you} \\$$
Answered by mr W last updated on 05/Feb/21
$$\int_{\pi} ^{\frac{\mathrm{4}\pi}{\mathrm{3}}} {f}\left({x}+\pi\right){dx} \\$$$$=\int_{\pi} ^{\frac{\mathrm{4}\pi}{\mathrm{3}}} {f}\left({x}+\pi\right){d}\left({x}+\pi\right) \\$$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{3}}} {f}\left({x}\right){dx} \\$$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} {f}\left({x}\right){dx}+\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} {f}\left({x}\right){dx} \\$$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} {f}\left({x}\right){dx}+\int_{\frac{\pi}{\mathrm{6}}} ^{\frac{\pi}{\mathrm{3}}} {f}\left({x}−\frac{\pi}{\mathrm{6}}\right){d}\left({x}−\frac{\pi}{\mathrm{6}}\right) \\$$$$=\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} {f}\left({x}\right){dx}+\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} {f}\left({x}\right){dx} \\$$$$=\mathrm{2}\int_{\mathrm{0}} ^{\frac{\pi}{\mathrm{6}}} {f}\left({x}\right){dx} \\$$$$=\mathrm{2}{T} \\$$
Commented by benjo_mathlover last updated on 05/Feb/21
$$\mathrm{thank}\:\mathrm{you} \\$$