Integral involving sine and root

I have found a way of solving it but I don't know if you will be able to follow, because I don't know what integration techniques you have learned. It's perhaps a bit over-complicated, as well. But I did the integration in Mathematica, and what I got reminded me very much of a Gaussian integral.

If you first do a substitution to u = x^1/2 you get an integral over something like sin(a u²). By symmetry you can extend this integral to the entire real axis. Then note that what you want is precisely the imaginary part of e^{i a u2}. Then you can use the result that the integral of exp(-a y²) over the real line is sqrt(pi / a) as long as Re(a) > 0 (technically, you will get Re(a) = 0 so you might need to take some limit, or perform a Wick rotation to make things rigorous) and you will get your answer.

Anyone for an easier approach?