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

4.25 Combinations of logarithms and powers

4.251

1 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{x^{\mu -1}ln\text{?}x}{\beta +x}\text{?}\text{d}x=\frac{\pi {\beta }^{\mu -1}}{sin\text{?}\mu \pi }}(ln\text{?}\beta -\pi \cot \text{?}\mu \pi )\hfill & [|\arg \text{?}\beta |\text{?}<\pi ,\text{?}0<\mathrm{Re}\text{?}\mu <1]\hfill \end{array}$

  BI (135)(1)

2. 

$\begin{array}{ll}{\displaystyle {\int }_0^{\infty }\frac{x^{\mu -1}ln\text{?}x}{a-x}\text{d}x=\pi a^{\mu -1}\text{?}\left(\cot \text{?}\mu \pi \text{?}ln\text{?}a-\frac{\pi }{{sin}^2\mu \pi }\right)}\hfill & [a>\text{?}0,\text{?}0<\mathrm{Re}\text{?}\mu <1\hfill \end{array}]$

  ET I 314(5)

3.¹⁰ 

$\begin{array}{ll}{\displaystyle {\int }_0^1\frac{x^{\mu -1}ln\text{?}x}{x+1}\text{d}x=\beta \text{'}(\mu )}\hfill & [\mathrm{Re}\text{?}\mu >0]\hfill \end{array}$

  GW (324)(6), ET I 314(3)

4. 

$\begin{array}{ll}{\displaystyle {\int }_0^1\frac{x^{\mu -1}ln\text{?}x}{x-1}\text{d}x=\psi \text{'}(\mu )=-\zeta (2,\mu )}\hfill & [\mathrm{Re}\text{?}\mu >0]\hfill \end{array}$

  BI (108)(8)

5.¹¹ 

${\displaystyle {\int }_0^{1}ln}$