$$$\csc^{2}{\left(x \right)} - 1$$$ 的積分

求$$$\int \left(\csc^{2}{\left(x \right)} - 1\right)\, dx$$$。

逐項積分：

$${\color{red}{\int{\left(\csc^{2}{\left(x \right)} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} + \int{\csc^{2}{\left(x \right)} d x}\right)}}$$

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dx = c x$$$：

$$\int{\csc^{2}{\left(x \right)} d x} - {\color{red}{\int{1 d x}}} = \int{\csc^{2}{\left(x \right)} d x} - {\color{red}{x}}$$

$$$\csc^{2}{\left(x \right)}$$$ 的積分是 $$$\int{\csc^{2}{\left(x \right)} d x} = - \cot{\left(x \right)}$$$：

$$- x + {\color{red}{\int{\csc^{2}{\left(x \right)} d x}}} = - x + {\color{red}{\left(- \cot{\left(x \right)}\right)}}$$

因此，

$$\int{\left(\csc^{2}{\left(x \right)} - 1\right)d x} = - x - \cot{\left(x \right)}$$

加上積分常數：

$$\int{\left(\csc^{2}{\left(x \right)} - 1\right)d x} = - x - \cot{\left(x \right)}+C$$

$$$\int \left(\csc^{2}{\left(x \right)} - 1\right)\, dx = \left(- x - \cot{\left(x \right)}\right) + C$$$A