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

4.38–4.41 Logarithms and trigonometric functions

4.381

1. 

$\begin{array}{ll}{{\displaystyle \int }}_0^{1}ln\text{?}x\text{?}sin\text{?}ax\text{?}\text{d}x=-\frac{1}{a}[C+ln\text{?}a-ci\text{?}(a)]\hfill & [a>0]\hfill \end{array}$

  GW (338)(2a)

2. 

$\begin{array}{ll}{\displaystyle {\int }_0^{1}ln}\text{?}x\text{?}cos\text{?}ax\text{?}\text{d}x=-\frac{1}{a}\left[\text{si}(a)+\frac{\pi }{2}\right]\hfill & [a>0]\hfill \end{array}$

  BI (284)(2)

3. 

${\displaystyle {\int }_0^{2\pi }ln}\text{?}x\text{?}sin\text{?}nx\text{?}\text{d}x=-\frac{1}{n}\left[C+ln(2n\pi )-\text{ci}(2n\pi )\right]$

  GW (338)(1a)

4. 

${\displaystyle {\int }_0^{2\pi }ln}\text{?}x\text{?}cos\text{?}nx\text{?}\text{d}x=-\frac{1}{n}\left[\text{si}(2n\pi )+\frac{\pi }{2}\right]$

  GW (338)(1b)

4.382

1.¹² 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }ln}\left|\frac{x+a}{x-a}\right|sin\text{?}bx\text{?}\text{d}x=\frac{\pi }{b}\text{?}sin\text{?}ab\text{?}\hfill & [a>0,\text{?}b>0]\hfill \end{array}$