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## 4.2–4.4 Logarithmic Functions

### 4.21 Logarithmic functions

4.211

1.

${\int }_{e}^{\infty }\frac{\text{d}x}{ln\frac{1}{x}}=-\infty$

BI (33)(9)

2.

${\int }_{e}^{u}\frac{\text{d}x}{lnx}=\text{li}\text{?}u$

FI III 653, FI II 606

4.212

1.

$\begin{array}{ll}{\int }_{0}^{1}\frac{\text{d}x}{a+lnx}={e}^{-a}\text{Ei}\left(a\right)\hfill & \left[a>0\right]\hfill \end{array}$

BI (31)(4)

2.

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

BI (31)(5)

3.7

$\begin{array}{ll}{\int }_{0}^{1}\frac{\text{d}x}{{\left(a+lnx\right)}^{2}}=-\frac{1}{a}+{e}^{-a}\text{Ei}\left(a\right)\hfill & \left[a\ge 0\right]\hfill \end{array}$

BI (31)(14)

4.

$\begin{array}{ll}{\int }_{0}^{1}\frac{\text{d}x}{{\left(a-lnx\right)}^{2}}=\frac{1}{a}+{e}^{a}\text{Ei}\left(-a\right)\hfill & \left[a>0\right]\hfill \end{array}$

BI (31)(16)

5.8

$\begin{array}{l}{\int }_{0}^{1}\frac{lnx\text{d}x}{}\hfill \end{array}$

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