Question Number 134672 by mnjuly1970 last updated on 06/Mar/21
![....nice calculus... prove that : (√(5π))≤ ∫_0 ^( (π/3)) ((9(√3))/(2π))(√(8sin(x)−sin(2x))) dx≤(√(6π)) ...m.n....](https://www.tinkutara.com/question/Q134672.png)
$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:….{nice}\:\:\:\:{calculus}… \\ $$$$\:\:\:\:\:{prove}\:\:{that}\:: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\sqrt{\mathrm{5}\pi}\leqslant\:\:\:\int_{\mathrm{0}} ^{\:\frac{\pi}{\mathrm{3}}} \frac{\mathrm{9}\sqrt{\mathrm{3}}}{\mathrm{2}\pi}\sqrt{\mathrm{8}{sin}\left({x}\right)−{sin}\left(\mathrm{2}{x}\right)}\:{dx}\leqslant\sqrt{\mathrm{6}\pi} \\ $$$$\:\:\:\:\:\:\:\:…{m}.{n}…. \\ $$