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nice-calculus-if-n-2-and-P-n-n-1-n-1-sin-kpi-n-find-lim-n-nP-n-2-pi-6-pi-3-cos-3x-sin-n-x-dx-




Question Number 135034 by mnjuly1970 last updated on 09/Mar/21
         ...nice   calculus        if  n≥2   and   P_n =Π_(n=1) ^(n−1) sin(((kπ)/n))       find :: lim_(n→∞) ((nP_n )/2) ∫_(π/6) ^( (π/3)) ((cos(3x))/(sin^n (x)))dx
$$\:\:\:\:\:\:\:\:\:…{nice}\:\:\:{calculus}\:\: \\ $$$$\:\:\:\:{if}\:\:{n}\geqslant\mathrm{2}\:\:\:{and}\:\:\:{P}_{{n}} =\underset{{n}=\mathrm{1}} {\overset{{n}−\mathrm{1}} {\prod}}{sin}\left(\frac{{k}\pi}{{n}}\right) \\ $$$$\:\:\:\:\:{find}\:::\:{lim}_{{n}\rightarrow\infty} \frac{{nP}_{{n}} }{\mathrm{2}}\:\int_{\frac{\pi}{\mathrm{6}}} ^{\:\frac{\pi}{\mathrm{3}}} \frac{{cos}\left(\mathrm{3}{x}\right)}{{sin}^{{n}} \left({x}\right)}{dx} \\ $$

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