I need an approach to analytically evaluating this limit:
$$\lim_{n\rightarrow\infty} \int_{-\pi}^\pi x^2 \frac{\sin(2nx)}{\sin x} dx$$
Numerically, I see that the answer is $-\pi^3$. Similarly, if I replace $x^2$ with $x^4$, I get $-\pi^5$. I vaguely recall seeing this result obtained analytically and not necessarily using advanced ideas, but I just can't remember any details. I know the fraction in the integrand relates to Chebyshev polynomials of the second kind. Thoughts anyone? Thanks!