I believe this integral $\int_0^a \frac{\cos(ux)}{\sqrt{a^2-x^2}}\mathrm dx$
can not be computed exactly. However is there a method or transformation to express this integral in terms of the cosine integral or similar? I am referring to the integrals here.
$a$ is real number; with the change of variable this integral becomes
$ \int_0^a\cos(u\sin t) \ \mathrm dt $ with $ x=a\sin t, $ So, the new integral is $ \int_0^{\pi /2}\cos(ua\sin t) \ \mathrm dt $