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Question Number 131496 by benjo_mathlover last updated on 05/Feb/21

$$\mathrm{Given}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}(\mathrm{x}+\frac{\pi }{6})\mathrm{\forall }\mathrm{x}\in \mathbb{R}$$$$\mathrm{if}{\int }_0^{\frac{\pi }{6}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{T}\mathrm{find}\mathrm{the}\mathrm{value}$$$$\mathrm{of}{\int }_{\pi }^{\frac{4\pi }{3}}\mathrm{f}(\mathrm{x}+\pi )\mathrm{dx}.$$$$\mathrm{nice}\mathrm{integral}$$

Answered by talminator2856791 last updated on 05/Feb/21

$$2\mathrm{T}$$

Commented by benjo_mathlover last updated on 05/Feb/21

$$\mathrm{why}?$$

Answered by EDWIN88 last updated on 05/Feb/21

$$\iff {\int }_{\pi }^{\frac{4}{3}\pi }\mathrm{f}(\mathrm{x}+\pi )\mathrm{dx}={\int }_{\pi -\pi }^{\frac{4\pi }{3}-\pi }\mathrm{f}(\mathrm{x}+\pi -\pi )\mathrm{dx}=$$$${\int }_0^{\frac{\pi }{3}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}={\int }_0^{0+2\left(\frac{\pi }{6}\right)}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=2{\int }_0^{\frac{\pi }{6}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}$$$$=2\mathrm{T}$$$$(\mathrm{by}\mathrm{theorem}\mathrm{If}{\int }_0^{\mathrm{P}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{G}\mathrm{then}{\int }_0^{\mathrm{n}.\mathrm{P}}\mathrm{f}\left(\mathrm{x}\right)\mathrm{dx}=\mathrm{n}.\mathrm{G})$$$$\mathrm{for}\mathrm{f}\left(\mathrm{x}\right)=\mathrm{f}(\mathrm{x}+\mathrm{P})\mathrm{\forall }\mathrm{x}\in \mathbb{R}$$

Commented by benjo_mathlover last updated on 05/Feb/21

$$\mathrm{thank}\mathrm{you}$$

Answered by mr W last updated on 05/Feb/21

$${\int }_{\pi }^{\frac{4\pi }{3}}f(x+\pi )dx$$$$={\int }_{\pi }^{\frac{4\pi }{3}}f(x+\pi )d(x+\pi )$$$$={\int }_0^{\frac{\pi }{3}}f\left(x\right)dx$$$$={\int }_0^{\frac{\pi }{6}}f\left(x\right)dx+{\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}f\left(x\right)dx$$$$={\int }_0^{\frac{\pi }{6}}f\left(x\right)dx+{\int }_{\frac{\pi }{6}}^{\frac{\pi }{3}}f(x-\frac{\pi }{6})d(x-\frac{\pi }{6})$$$$={\int }_0^{\frac{\pi }{6}}f\left(x\right)dx+{\int }_0^{\frac{\pi }{6}}f\left(x\right)dx$$$$=2{\int }_0^{\frac{\pi }{6}}f\left(x\right)dx$$$$=2T$$

Commented by benjo_mathlover last updated on 05/Feb/21

$$\mathrm{thank}\mathrm{you}$$

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