How can we calculate
$$
\int\_{0}^{\infty}{\sin\left(x\right)\sin\left(2x\right)\sin\left(3x\right)\ldots
\sin\left(nx\right)\sin\left(n^{2}x\right) \over x^{n + 1}}\,\mathrm{d}x ?
$$

I believe that we can use the Dirichlet integral

$$
\int\_{0}^{\infty}{\sin\left(x\right) \over x}\,\mathrm{d}x =
{\pi \over 2}
$$

But how do we split the integrand?