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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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