$$$\frac{1}{\sin^{2}{\left(x \right)}}$$$ 的積分

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

將被積函數改寫為以餘割函數表示:

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

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

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

因此，

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

加上積分常數：

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

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