Evaluation of integral $\int_{0}^{\infty} \frac{\sqrt{k}}{(k+\frac{1}{2})^{n+2}} J_{0}(2k\sin\theta) \sin{(\tau\sqrt{k})}\mathrm{d}k$

Question

How to compute the integral that showed in the title. $J_0$ is the Bessel function. Parameters $\tau$ is a positive real number and $\theta\in[0,2\pi]$, n is a positive integer. Any help on it is much appreciated. Many thanks in advance. $$ \int_{0}^{\infty} \frac{\sqrt{k}}{(k+\frac{1}{2})^{n+2}} J_{0}(2k\sin\theta) \sin{(\tau\sqrt{k})}\mathrm{d}k = \int_{0}^{\infty} \frac{2k^2}{(k^2+\frac{1}{2})^{n+2}} J_{0}(2k^2\sin\theta) \sin{(\tau k)}\mathrm{d}k $$