# 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 '10 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 '10 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 $0<m<1$. That's the elementary end of the story. –  Wadim Zudilin Oct 15 '10 at 10:23
Though not relevant here, Carlson's theorem (en.wikipedia.org/wiki/Carlson's_theorem) can sometimes be used to extend a formula for $f(n)$, where $n$ is an integer, to all complex numbers. A good example is the evaluation of Selberg's integral (e.g., page 2 of ms.unimelb.edu.au/~warnaar/pubs/Selberg_draft.pdf). –  Richard Stanley Oct 15 '10 at 17:19
@Wadim would you kindly name the technique that I should look at. I haven't studied fractional calculus yet but I'm willing to. However I'd like to solve integrate this expression as soon as possible in order to continue writing my thesis. Thanks. –  Olumide Oct 15 '10 at 23:06