$$$\frac{e}{\ln\left(x\right)}$$$ 的積分

求$$$\int \frac{e}{\ln\left(x\right)}\, dx$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=e$$$ 與 $$$f{\left(x \right)} = \frac{1}{\ln{\left(x \right)}}$$$：

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

此積分（對數積分）不存在閉式表示：

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

因此，

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

加上積分常數：

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

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