  For $\alpha>\beta-1>0$, we obtain two sided inequalities for the moment
integral $I(\alpha,\beta)= \int_{\mathbb{R}} |x|^{-\beta}|\sin x|^{\alpha}dx$.
These are then used to give the exact asymptotic behavior of the integral as
$\alpha \to \infty$. The case $I(\alpha,\alpha)$ corresponds to the asymptotics
of Ball's inequality, and $I(\alpha,[\alpha]-1)$ corresponds to a kind of novel
"oscillatory" behavior.
