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$\begin{array}{ll}{\int }_{0}^{1}ln\text{?}\left(-ln\text{?}x\right)\frac{\text{d}x}{\left(1+{x}^{2}\right)\sqrt{-ln\text{?}x}}\hfill & ={\int }_{1}^{\infty }ln\text{?}ln\text{?}x\frac{\text{d}x}{\left(1+{x}^{2}\right)\sqrt{ln\text{?}x}}\hfill \\ =\sqrt{\pi }\sum _{k=0}^{\infty }\frac{{\left(-1\right)}^{k+1}}{\sqrt{2k+1}}\left[ln\text{?}\left(2k+1\right)+2ln\text{?}2+C\right]\hfill \end{array}$

BI (147)(4)

11.

$\begin{array}{ll}{\int }_{0}^{1}ln\text{?}\left(-ln\text{?}x\right)\frac{{x}^{\mu -1}\text{d}x}{\sqrt{-ln\text{?}x}}=-\left(C+ln\text{?}4\mu \right)\sqrt{\frac{\pi }{\mu }}\hfill & \left[\mathrm{Re}\text{?}\mu >0\right]\hfill \end{array}$

BI (147)(3)

12.

${\int }_{0}^{1}ln\text{?}\left(-ln\text{?}x\right){\left(-ln\text{?}x\right)}^{u-1}{x}^{v-1}\text{d}x=\frac{1}{{v}^{\mu }}\mathrm{\Gamma }\left(\mu \right)\left[\psi \left(\mu \right)-ln\text{?}\left(v\right)\right]$

BI (147)(2)

$[Re?μ>0,?Re?v>0]$

4.326

1.

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

BI (107)(23)

2.

$\begin{array}{l}{\int }_{0}^{1/e}ln\hfill \end{array}$

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