$$$- \frac{\sin{\left(x \right)}}{x}$$$ 的積分
您的輸入
求$$$\int \left(- \frac{\sin{\left(x \right)}}{x}\right)\, dx$$$。
解答
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=-1$$$ 與 $$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{x}$$$：
$${\color{red}{\int{\left(- \frac{\sin{\left(x \right)}}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{\sin{\left(x \right)}}{x} d x}\right)}}$$
此積分（正弦積分）不存在閉式表示：
$$- {\color{red}{\int{\frac{\sin{\left(x \right)}}{x} d x}}} = - {\color{red}{\operatorname{Si}{\left(x \right)}}}$$
因此，
$$\int{\left(- \frac{\sin{\left(x \right)}}{x}\right)d x} = - \operatorname{Si}{\left(x \right)}$$
加上積分常數：
$$\int{\left(- \frac{\sin{\left(x \right)}}{x}\right)d x} = - \operatorname{Si}{\left(x \right)}+C$$
答案
$$$\int \left(- \frac{\sin{\left(x \right)}}{x}\right)\, dx = - \operatorname{Si}{\left(x \right)} + C$$$A