Integral of $$$\frac{e}{\ln\left(x\right)}$$$

Find $$$\int \frac{e}{\ln\left(x\right)}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=e$$$ and $$$f{\left(x \right)} = \frac{1}{\ln{\left(x \right)}}$$$:

$${\color{red}{\int{\frac{e}{\ln{\left(x \right)}} d x}}} = {\color{red}{e \int{\frac{1}{\ln{\left(x \right)}} d x}}}$$

This integral (Logarithmic Integral) does not have a closed form:

$$e {\color{red}{\int{\frac{1}{\ln{\left(x \right)}} d x}}} = e {\color{red}{\operatorname{li}{\left(x \right)}}}$$

Therefore,

$$\int{\frac{e}{\ln{\left(x \right)}} d x} = e \operatorname{li}{\left(x \right)}$$

Add the constant of integration:

$$\int{\frac{e}{\ln{\left(x \right)}} d x} = e \operatorname{li}{\left(x \right)}+C$$

$$$\int \frac{e}{\ln\left(x\right)}\, dx = e \operatorname{li}{\left(x \right)} + C$$$A