$\displaystyle{\large%
{\cal I}_{mn}
\equiv
\int_{0}^{\pi}
\left\lbrack{\sin\left(nx\right) \\sin\left(x\right)}\right\rbrack^{m}\,{\rm d}x
\ = \
?\,,
\quad m, n = 0,1,2,\ldots}$
$$
\lim_{m \to 0}\lim_{n \to 0}{\cal I}_{mn} = {\cal I}_{10} = 0\,,
\qquad
\lim_{n \to 0}\lim_{m \to 0}{\cal I}_{mn} = {\cal I}_{01} = {\cal I}_{11} = \pi
$$
Consideremos $m > 1$$n > 1$:
\begin{align}
{\cal I}_{mn}
&=
{1 \over 2}\int_{0}^{2\pi}{\rm e}^{-{\rm i}m\left(n - 1\right)x/2}
\left(
{{\rm e}^{{\rm i}nx} - 1
\over
{\rm e}^{{\rm i}x} - 1}
\right)^{m}\,{\rm d}x
=
{1 \over 2}\oint_{\left\vert z\right\vert = 1}{1 \over z^{m\left(n - 1\right)/2}}
\left(%
{z^{n} - 1 \over z - 1}
\right)^{m}\,{{\rm d}z \over {\rm i}z}
\\[3mm]&=
\pi\oint_{\left\vert z\right\vert = 1}{{\rm d}z \over 2\pi{\rm i}}\,{1 \over z}\,
{1 \over z^{m\left(n - 1\right)/2}}
\left({1 - z^{n} \over 1 - z}\right)^{m}
\end{align}
A continuación, $\left.{\cal I}_{mn}\vphantom{\Large A}\right\vert_{m\ >\ 1 \atop n\ >\ 1} = 0$
siempre que $\phantom{A}m\left(n - 1\right)\phantom{A}$ es impar. Que significa
$\left.{\cal I}_{mn}\vphantom{\Large A}\right\vert_{m > 1 \atop n > 1} = 0$ al $m$ es impar y $n$ es incluso.
\begin{align}
\left(1 - z^{n} \over 1 - z\right)^{m}
&=
\sum_{k = 0}^{m}{m \choose k}\left(-1\right)^{k}z^{nk}
\sum_{k' = 0}^{\infty}{m + k'\choose k'}z^{k'}
\sum_{\ell = 0}^{\infty}\delta_{\ell, nk + k'}
\\[3mm]&=
\sum_{\ell = 0}^{\infty}z^{\ell}
\sum_{k = 0}^{\leq\ M \atop}\left(-1\right)^{k}{m \choose k}
{m + \ell - nk \choose \ell - nk}
\\[3mm]&=
\sum_{\ell = 0}^{\infty}z^{\ell}
\sum_{k = 0}^{\leq\ M \atop}\left(-1\right)^{k}
{\left(m + \ell -nk\right)!
\over
k!\left(m - k\right)!\left(\ell - nk\right)!}\,,
\quad
M \equiv \min\left\lbrace m, {\ell \over n}\right\rbrace
\\[1cm]&
\end{align}
\begin{align}
\int_{0}^{\pi}
\left\lbrack{\sin\left(nx\right) \over \sin\left(x\right)}\right\rbrack^{m}\,{\rm d}x
&=
\pi\sum_{k = 0}^{\leq\ M\atop}\left(-1\right)^{k}
{\left\lbrack m\left(n + 1\right)/2 -nk\right\rbrack!
\over
k!\left(m - k\right)!\left\lbrack m\left(n - 1\right)/2 - nk\right\rbrack!}
\\[2mm]&
\mbox{where}\quad
M \equiv \min\left\lbrace m, {1 \over 2}\,m\left(1 - {1 \over n}\right)\right\rbrace
=
{1 \over 2}\,m\left(1 - {1 \over n}\right)
\end{align}
Ejemplos:
\begin{align}
{\cal I}_{1n}
&=
\int_{0}^{\pi}{\sin\left(nx\right) \over \sin\left(x\right)}\,{\rm d}x
=
\pi
{\left\lbrack \left(n + 1\right)/2\right\rbrack!
\over
\left\lbrack \left(n - 1\right)/2\right\rbrack!}
=
{1 \over 2}\,\left(n + 1\right)\pi\,,
\qquad
n = 3, 5, 7, \ldots
\\[3mm]
{\cal I}_{2n}
&=
\int_{0}^{\pi}\left\lbrack{\sin\left(nx\right) \over \sin\left(x\right)}\right\rbrack^{2}\,{\rm d}x
=
\pi\,{\left(n + 1\right)! \over \left(n - 1\right)!}
=
n\left(n + 1\right)\pi\,,
\qquad
n = 2, 3, 4, \ldots
\\[6mm]
{\cal I}_{3n}
&=
\int_{0}^{\pi}\left\lbrack{\sin\left(nx\right) \over \sin\left(x\right)}\right\rbrack^{3}\,{\rm d}x
=
\pi\,
{\left\lbrack 3\left(n + 1\right)/2\right\rbrack!
\over
3!\left\lbrack 3\left(n - 1\right)/2\right\rbrack!}
-
\pi\,
{\left\lbrack 3\left(n + 1\right)/2 - n\right\rbrack!
\over
2!\left\lbrack 3\left(n - 1\right)/2 -n\right\rbrack!}
\\[3mm]&=
{\pi \over 6}\,{3n + 3 \over 2}\,{3n + 1 \over 2}
-
{\pi \over 2}\,{n + 3 \over 2}\,{n + 1 \over 2}
=
{1 \over 4}\,\left(n^{2} - 1\right)\pi\,,
\qquad
n = 3, 5, 7, \ldots
\end{align}
