complex analysis - Various evalutions of $int_0^infty sin x sin sqrt{x} ,dx$

I'm looking for various ways to evaluate the integral:
$$\int_0^\infty \sin x\sin \sqrt{x}\,dx$$

I'm mainly interested in complex analysis. I can think of a wedge -shaped contour of angle $\pi/4$ but I'm having trouble constructing the integrand properly. Perhaps we have to take into account that the root here may cause some trouble and define a function having a branch and this complicates things.

I know two solutions using real analysis. One uses Laplace transformations and the other using only elementary tools plus the known results of the Fresnel Integrals.

Can someone help me with the contour integration?
Thank you!