How to see this integral is bounded for every positive integer $n$? $$\int_{-\pi}^{\pi} \frac{1}{n} \frac{\sin^2(nx/2)}{\sin^2(x/2)} dx$$ Since there is a singular point at $x=0$, I cannot argue the behavior near the origin. Any help is appreciated.
2026-03-29 10:14:32.1774779272
How to see this integral is bounded for every positive integer n?
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Using $$\frac{\sin^2(nx/2)}{\sin^2(x/2)}=\frac{1-e^{inx}}{1-e^{ix}}\frac{1-e^{-inx}}{1-e^{-ix}}=\sum_{j,k=0}^{n-1}e^{i(j-k)x},$$ the integral is easily evaluated exactly (it is equal to $2\pi$).