$$$\frac{e}{\ln\left(x\right)}$$$の積分

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

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=e$$$ と $$$f{\left(x \right)} = \frac{1}{\ln{\left(x \right)}}$$$ に対して適用する:

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

この積分（対数積分）には閉形式はありません:

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

したがって、

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

積分定数を加える:

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

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