In summary, the conversation discusses the integration of sin(ax)/sqrt(x) using different techniques. One approach involves integration by parts, but it does not provide a solution. Another approach uses a substitution and extends the integral to the entire real axis to obtain the imaginary part of ei au^2. This method ultimately leads to the result that the integral of exp(-a y^2) over the real line is sqrt(pi/a) as long as Re(a) > 0. The conversation concludes with an inquiry for a simpler approach.
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 = x1/2 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 ei 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.