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: $$\frac{1}{2}a^{2\nu-2\mu}\frac{\Gamma(1+\nu)\Gamma(\mu-\nu)}{\Gamma(\mu+1)}y \:_1\text{F}_2(\nu+1;\nu+1-\mu,3/2;a^2y^2/4)\:+\:4^{\nu-\mu-1}\sqrt{\pi}\frac{\Gamma(\nu-\mu)}{\Gamma(\mu-\nu+3/2)}y^{2\mu-2\nu+1}\:_1\text{F}_2(\mu+1;\mu-\nu+3/2,\mu-\nu+1;a^2y^2/4). $$ In particular, the integral $\int_{0}^{\infty} \frac{\sqrt{x}\sin(x)}{1+x^2} dx$ is expressed in terms of the error functions: $$\frac{\pi}{2\sqrt{2}\:e}\left(-e^2\text{erfc}(1)+\text{erf}(1) +1\right).$$

However, no derivation is provided there. I am seeking a derivation of the above expressions.

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