Evaluation of definite integral in terms of Bessel function

Question

Can I express the integral $\int_0^1[\cos (xt)/(1-t^2)]dt$ in terms of Bessel Polynomial? I tried by putting $t=\sin \theta$ and used the integral representation of Bessel's polynomial $J_n(x)=(1/\pi)\int_0^\pi \cos(n\theta-x\sin \theta)d\theta$. I expect the answer may be $J_0(x)(\pi/2)$. But I did not get it even now. Help is solicited.

Answer 1

Due to issues relating to convergence, the only values of x for which the integral converges are

of the form $x=(2k+1)~\dfrac\pi2$ , and the result is $I_{2k+1}=\dfrac\pi4\sqrt{|2k+1|}\cdot J^{(1,0)}\bigg(-\dfrac12~,~|2k+1|\dfrac\pi2\bigg)$,

for all $k\in\mathbb Z$.