In this answer on math.stackexchange.com the Fourier Sine Transform of $ x^{2\nu}(x^2+a^2)^{-\mu-1}$ is given in terms of the generalized hypergeometric function. In particular, the integral $ \int_{0}^{\infty} \frac{\sqrt{x}\sin(x)}{1+x^2} dx$ is expressed in terms of the error functions. However, no derivation is provided there. I am seeking a derivation of the above expressions.Read more