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

3.78–3.81 Rational functions of x and of trigonometric functions

3.781

1. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\left(\frac{sinx}{x}-\frac{1}{1+x}\right)\frac{\text{d}x}{x}=1-C}\hfill & \left(\text{cf}\text{.}3.784\text{4}\text{and}3.781\text{2}\right)\hfill \end{array}$

  BI (173)(7)

2. 

${\displaystyle {\int }_0^{\infty }\left(cosx-\frac{1}{1+x}\right)\frac{\text{d}x}{x}=-C}$

  BI (173)(8)

3.782

1. 

$\begin{array}{ll}{\displaystyle {\int }_0^u\frac{1-cosx}{x}\text{d}x-}{\displaystyle {\int }_u^{\infty }\frac{cosx}{x}\text{d}x=C+\ln u}\hfill & \left[u>0\right]\hfill \end{array}$

  GW (333)(31)

2. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{1-cosax}{x^2}\text{d}x}=\frac{a\pi }{2}\hfill & \left[a\ge 0\right]\hfill \end{array}$

  BI (158)(1)

3.¹² 

$\begin{array}{ll}{\displaystyle {\int }_{-\infty }^{\infty }\frac{1-cosx}{x(x-b)}\text{d}x=-\frac{\pi i}{b}\left(e^{ib}-1\right)}\hfill & \left[\begin{array}{ll}\text{Im}b\ne 0,\hfill & \text{Re}{b}^2\le 0\hfill \end{array}\right]\hfill \end{array}$

  ET II 253(48)

3.783 ...