Question Number 82223 by jagoll last updated on 19/Feb/20
$${given}\:{f}\left({x}\right)\:=\:{f}\left({x}+\frac{\pi}{\mathrm{6}}\right)\:,\:\forall{x}\in\:\mathbb{R} \\ $$$${if}\:\underset{\mathrm{0}} {\overset{\frac{\pi}{\mathrm{6}}} {\int}}\:{f}\left({x}\right)\:{dx}\:=\:{T}\: \\ $$$${then}\:\underset{\pi} {\overset{\frac{\mathrm{7}\pi}{\mathrm{3}}} {\int}}\:{f}\left({x}+\pi\right)\:{dx}\:=\:? \\ $$
Commented by john santu last updated on 19/Feb/20
$$\Rightarrow\:\underset{\pi} {\overset{\pi\:+\:\mathrm{8}\left(\frac{\pi}{\mathrm{6}}\right)} {\int}}{f}\left({x}+\mathrm{6}.\frac{\pi}{\mathrm{6}}\right)\:{dx}\:=\:\mathrm{8}{T} \\ $$