### 4.25 Combinations of logarithms and powers

4.251

1

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

BI (135)(1)

2.

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

ET I 314(5)

3.10

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

GW (324)(6), ET I 314(3)

4.

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

BI (108)(8)

5.11

${\int }_{0}^{1}ln$

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