I'm looking to integrate several functions having the form

$\int_0^T \frac{ sin(\omega \tau) }{\omega} \tau^{2H} d\tau$

where $2H \ge 0$ but may not be an integer. I'd like to know if the machinery of fractional calculus (e.g. fractional integration by parts) can be used to integrate such an expression.

Thanks.

(PS: I'm not quite sure to which category/tag this question belongs.)