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