## Integrating the product of two functions one of which has a positive non-integer power

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.)

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 Is it a joke? The antiderivative of $\sin(a\tau)\tau^m$ is elementary for $m$ nonnegative integers only. Otherwise, integrate the corresponding power series. This isn't appropriate for MO. – Wadim Zudilin Oct 15 2010 at 9:20 No it is not a joke. The problem seemed much harder to me at first and still does a bit. Specifically the possibility of antiderivatives of non-integer powers is still very new to me. Nevertheless, thanks for alerting me to both possibilities. – Olumide Oct 15 2010 at 9:51 You are welcome, Olumide. I might be too strong in my opinion but it's not only the author who judges how hard is a problem. You can use integration by parts (which you can count as "fractional calculus") to reduce your problem to the case \$0