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Can the integral $$\int_{0}^{x} \frac{\cos(ut)}{\sqrt{x-t}}dt $$ be expressed in terms of elemental functions or in terms of the sine and cosine integrals ? if possible i would need a hint thanks.

From the fractional calculus i guess this integral is the half derivative of the sine function (i think so) $ \sqrt \pi \frac{d^{1/2}}{dx^{1/2}}\sin(ux) $ or similar

of course i could expand the cosine into power series and then take the term by term integration but i would like if possible a closed expression for my integral

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    Dear @Jose check if I editted right? :)2012-09-13
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    Does it help to use $v = \frac{t}{x}$?2012-09-13
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    aha.. yes Babak nice edit2012-09-13
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    The title does not correspond to the question.2012-09-13
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    I just wanted to emphasize that. (+1)2012-09-13
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    I may suppose it has some connections with Fresnel integrals.2012-09-13
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    oh sorry i wanted to write 'evaluation' instead of convergence :D ca someone edit the title ?2012-09-13
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    Maple evaluates this in terms of Fresnel integrals. So I guess it is not elementary.2012-09-13
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    Make the right substitution (and trig expansion), and end up with indefinite integrals of $\cos(y)/\sqrt{y}$ and $\sin(y)/\sqrt{y}$.2012-09-13

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