$$$- \frac{\sin{\left(x \right)}}{x}$$$の積分

$$$\int \left(- \frac{\sin{\left(x \right)}}{x}\right)\, dx$$$ 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、$$$c=-1$$$ と $$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{x}$$$ に対して適用する:

$${\color{red}{\int{\left(- \frac{\sin{\left(x \right)}}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{\sin{\left(x \right)}}{x} d x}\right)}}$$

この積分（正弦積分）には閉形式はありません:

$$- {\color{red}{\int{\frac{\sin{\left(x \right)}}{x} d x}}} = - {\color{red}{\operatorname{Si}{\left(x \right)}}}$$

したがって、

$$\int{\left(- \frac{\sin{\left(x \right)}}{x}\right)d x} = - \operatorname{Si}{\left(x \right)}$$

積分定数を加える:

$$\int{\left(- \frac{\sin{\left(x \right)}}{x}\right)d x} = - \operatorname{Si}{\left(x \right)}+C$$

$$$\int \left(- \frac{\sin{\left(x \right)}}{x}\right)\, dx = - \operatorname{Si}{\left(x \right)} + C$$$A