Evaluating $\int_0^\infty \frac{e^{-kx}\sin x}x\,\mathrm dx$ How to evaluate the following integral?
$$\int_0^\infty \frac{e^{-kx}\sin x}x\,\mathrm dx$$

One more option:$$\begin{align}\int_0^\infty\int_0^\infty e^{-(k+y)x}\sin x\mathrm{d}x\mathrm{d}y&=\Im\int_0^\infty\int_0^\infty e^{-(k+y-i)x}\mathrm{d}x\mathrm{d}y\\&=\int_0^\infty\tfrac{1}{(k+y)^2+1}\mathrm{d}y\\&=[\arctan(k+y)]_0^\infty\\&=\tfrac{\pi}{2}-\arctan k.\end{align}$$
