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

  • $\begingroup$ Dear @Jose check if I editted right? :) $\endgroup$ – mrs Sep 13 '12 at 10:20
  • $\begingroup$ Does it help to use $v = \frac{t}{x}$? $\endgroup$ – user 1591719 Sep 13 '12 at 10:33
  • $\begingroup$ aha.. yes Babak nice edit $\endgroup$ – Jose Garcia Sep 13 '12 at 10:46
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    $\begingroup$ The title does not correspond to the question. $\endgroup$ – Did Sep 13 '12 at 10:47
  • $\begingroup$ I just wanted to emphasize that. (+1) $\endgroup$ – user 1591719 Sep 13 '12 at 10:48

Let $t=x-y^2$. We then have $dt = -2ydy$. Hence, we get that \begin{align} I = \int_0^x \dfrac{\cos(ut)}{\sqrt{x-t}} dt & = \int_0^{\sqrt{x}} \dfrac{\cos(u(x-y^2))}{y} \cdot 2y dy\\ & = 2 \cos(ux) \int_0^{\sqrt{x}}\cos(uy^2)dy + 2 \sin(ux) \int_0^{\sqrt{x}}\sin(uy^2)dy\\ & = \dfrac{\sqrt{2 \pi}}{\sqrt{u}} \left(\cos(ux) C\left(\sqrt{\dfrac{2ux}{\pi}}\right) + \sin(ux) S\left(\sqrt{\dfrac{2ux}{\pi}}\right) \right) \end{align} where $C(z)$ and $S(z)$ are the Fresnel integrals.


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