Question Number 103515 by bemath last updated on 15/Jul/20
$${given}\:{f}\left({x}\right)\:=\:{f}\left({x}+\frac{\pi}{\mathrm{6}}\right)\:\forall{x}\in\mathbb{R} \\ $$$${if}\:\underset{\mathrm{0}} {\overset{\pi/\mathrm{6}} {\int}}{f}\left({x}\right){dx}\:=\:{T}\:{then}\:\underset{\pi} {\overset{\mathrm{7}\pi/\mathrm{3}} {\int}}{f}\left({x}+\pi\right) \\ $$$${dx}\:? \\ $$
Answered by bramlex last updated on 15/Jul/20
$$\Rightarrow\underset{\pi} {\overset{\mathrm{7}\pi/\mathrm{3}} {\int}}\mathrm{f}\left(\mathrm{x}+\pi\right)\:\mathrm{dx}\:=\:\underset{\mathrm{0}} {\overset{\mathrm{4}\pi/\mathrm{3}} {\int}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}\:= \\ $$$$\underset{\mathrm{0}} {\overset{\mathrm{8}\left(\frac{\pi}{\mathrm{6}}\right)} {\int}}\mathrm{f}\left(\mathrm{x}\right)\:\mathrm{dx}\:=\:\mathrm{8T}\:\blacksquare \\ $$