I have to evaluate the Fourier sine integral $$\int_0^\infty\frac{\sin(sx)}{x(a^2+x^2)}\mathrm dx$$
I tried using contour integration using
$$\int_0^∞\frac{\sin(sx)}{x(a^2+x^2)}\mathrm dx = \frac{1}{2}\int_c\frac{e^{isz}}{z(a^2+z^2)}\mathrm dz$$
Which on solving gives me $\frac{\pi i}{a^2}(1-\frac{e^{-as}}{2})$
I have used $ai$ and $0$ as my poles
But the final required solution is $$\frac{π}{2a^2}(1-e^{-as})$$
Where am I going wrong can someone help me