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4.38–4.41 Logarithms and trigonometric functions

4.381

1.

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

GW (338)(2a)

2.

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

BI (284)(2)

3.

${\int }_{0}^{2\pi }ln\text{?}x\text{?}sin\text{?}nx\text{?}\text{d}x=-\frac{1}{n}\left[C+ln\left(2n\pi \right)-\text{ci}\left(2n\pi \right)\right]$

GW (338)(1a)

4.

${\int }_{0}^{2\pi }ln\text{?}x\text{?}cos\text{?}nx\text{?}\text{d}x=-\frac{1}{n}\left[\text{si}\left(2n\pi \right)+\frac{\pi }{2}\right]$

GW (338)(1b)

4.382

1.12

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

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