How can I find the Fourier sine transform of $arctan(x/a) ;a>0$? I am solving it as, $$F_{s}\left[\arctan\left(\frac{x}{a}\right)\right]=\sqrt{\frac{2}{\pi}}\int_{0}^{\infty}\arctan\left(\frac{x}{a}\right)\sin(wx)dx$$ $$F_{s}\left[\arctan\left(\frac{x}{a}\right)\right]=\sqrt{\frac{2}{\pi}}\left[ \left|\arctan\left(\frac{x}{a}\right)\frac{\cos(wx)}{w}\right|^{\infty}_{0}-\int_{0}^{\infty}\frac{a}{a^2+x^2}\frac{\cos(ws)}{w}dx\right]$$
Since $\cos(\infty)$ is undefined, I am not able to evaluate the definite integral in the term inside the bracket.
How can I solve it? Should I go for different approach to find Fourier sine transform of $\arctan(x/a)$?
Please clarify my doubts!
Thank you!