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For given $q$ it is easy to compute this integral using integration by parts. For general integer(even) $q>0$, Mathematica gives the formula: $-Ei[-q,iX]+(ix)^{-1-q}q!$ where $Ei$ is exponential integral. Is exponential integral related with some known polynomials?

I also need a real and imaginary part of this expression.

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    I dont like neither of them :) If consider integrals $\int_{0}^{1}t^{q}\sin[Xt]dt $ and $\int_{0}^{1}t^{q}\cos[Xt]dt $ the result is generalized hypergeometric functions2011-06-21

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This is basically the same question as How to integrate $ \int x^n e^x dx$? plus a change of variables. The answer, for any nonnegative integer $q$, is $ \left(\frac{i}{X}\right)^{1+q} \left( (-1)^q q! (e^{-iX} - 1) + \sum_{j=1}^q (-1)^{q-j} \frac{q!}{j!} (-iX)^j e^{-iX} \right)$

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    this pol$y$nomial doesn't have a special name or properties, does it?2011-06-21