Table of Integrals, Series, and Products, 8th Edition

$\begin{array}{cc}{\displaystyle {\int }_0^{\infty }\left[\text{s}\text{i}\left(ax\right)+\frac{\pi }{2}\right]}cosbx\frac{x\mathrm{d}x}{x^2+c^2}=-\frac{\pi }{4c}\left\{e^{-bc}\left[\text{E}\text{i}\left(bc\right)-\text{E}\text{i}\left(-ac\right)\right]+e^{bc}\left[\text{E}\text{i}\left(-bc\right)-\text{E}\text{i}\left(-ac\right)\right]\right\}\text{?}\text{?}\left[0<b\le a,\text{?}\text{?}c>0\right]=\frac{\pi }{4c}{e}^{-bc}\left[\text{E}\text{i}\left(-ac\right)-\text{E}\text{i}\left(ac\right)\right]& \left[0<a\le b,\text{?}\text{?}c>0\right]\end{array}$

  BI (460)(2,5)

6.259

1. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\text{s}\text{i}\left(ax\right)}\text{?}\text{?}sinbx\frac{\mathrm{d}x}{x^2+c^2}& =\frac{\pi }{2c}\text{E}\text{i}\left(-ac\right)sinh\left(bc\right)\\ =\frac{\pi }{4c}\text{?}{e}^{-bc}\left[\text{E}\text{i}\left(-bc\right)+\text{E}\text{i}\left(bc\right)\text{?}\text{?}-\text{E}\text{i}\left(-ac\right)-\text{E}\text{i}\left(-ac\right)\text{?}\right]& \left[0<b\le a,\text{?}\text{?}c>0\right]\\ +\frac{\pi }{2c}\text{E}\text{i}\left(-bc\right)sinh\left(bc\right)& \left[0<a\le b,\text{?}\text{?}c>0\right]\end{array}$

  ET I 96(8)

2. 

$\begin{array}{lll}{\displaystyle {\int }_0^{\infty }\text{c}\text{i}\left(ax\right)}\text{?}\text{?}sinbx\frac{x\mathrm{d}x}{x^2+c^2}& =-\frac{\pi }{2}sinh\left(bc\right)\text{E}\text{i}\left(-ac\right)& \left[0<b\le a,\text{?}\text{?}c>0\right]\\ =-\frac{\pi }{2}sinh\left(bc\right)\text{E}\text{i}\left(-bc\right)+\frac{\pi }{4}{e}^{-bc}[\text{E}\text{i}\left(-bc\right)+\text{E}\text{i}\left(bc\right)\\ -\text{E}\text{i}\left(-ac\right)-\text{E}\text{i}\left(-ac\right)]& \left[0<a\le b,\text{?}\text{?}c>0\right]\end{array}$

  BI (460)(3)a, ...