### 3.78–3.81 Rational functions of x and of trigonometric functions

3.781

1.

$\begin{array}{ll}{\int }_{0}^{\infty }\left(\frac{sinx}{x}-\frac{1}{1+x}\right)\frac{\text{d}x}{x}=1-C\hfill & \left(\text{cf}\text{.}3.784\text{4}\text{and}3.781\text{2}\right)\hfill \end{array}$ BI (173)(7)

2.

${\int }_{0}^{\infty }\left(cosx-\frac{1}{1+x}\right)\frac{\text{d}x}{x}=-C$ BI (173)(8)

3.782

1.

$\begin{array}{ll}{\int }_{0}^{u}\frac{1-cosx}{x}\text{d}x-{\int }_{u}^{\infty }\frac{cosx}{x}\text{d}x=C+\mathrm{ln}u\hfill & \left[u>0\right]\hfill \end{array}$ GW (333)(31)

2.

$\begin{array}{ll}{\int }_{0}^{\infty }\frac{1-cosax}{{x}^{2}}\text{d}x=\frac{a\pi }{2}\hfill & \left[a\ge 0\right]\hfill \end{array}$ BI (158)(1)

3.12

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

ET II 253(48)

3.783 ...

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