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

3.72–3.74 Combinations of trigonometric and rational functions

3.721

1. 

${\displaystyle {\int }_0^{\infty }\frac{\text{sin}(ax)}{x}\text{d}x=\frac{\pi }{2}\text{sign}a}$

  FI II 645

2. 

${\displaystyle {\int }_1^{\infty }\frac{\text{sin}(ax)}{x}\text{d}x=-\text{si}(a)}$

  BI 203(5)

3.⁸ 

${\displaystyle {\int }_1^{\infty }\frac{\text{cos}\left(ax\right)}{x}\text{d}x=-\text{ci}(a)}$

  BI 203(5)

3.722

1. 

${\displaystyle {\int }_0^{\infty }\begin{array}{ll}\frac{\text{sin}(ax)}{x+b}\text{d}x=\text{ci}(ab)\text{sin}\left(ab\right)-\text{cos}\left(ab\right)\text{si}\left(ab\right)\hfill & [|\text{arg}b|<\pi ,a>0]\hfill \end{array}}$

  BI(16)(1), FI II 646a

2.¹¹ 

$\begin{array}{ll}{\displaystyle {\int }_{-\infty }^{\infty }\frac{\text{sin}(ax)}{x+b}\text{d}x=\pi e^{iab}}\hfill & [a>0,\text{Im}b>0]\hfill \end{array}$

3. 

$\begin{array}{l}{\displaystyle {\int }_0^{\infty }\frac{\text{cos}\left(ax\right)}{x+b}\text{d}x=-\text{sin}(a}\hfill \end{array}$