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

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

Expand the expression:

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

逐項積分：

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

將被積函數改寫為以餘割函數表示:

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

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

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

令 $$$u=\sin{\left(x \right)}$$$。

則 $$$du=\left(\sin{\left(x \right)}\right)^{\prime }dx = \cos{\left(x \right)} dx$$$ (步驟見»)，並可得 $$$\cos{\left(x \right)} dx = du$$$。

該積分可改寫為

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

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=-2$$$：

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

回顧一下 $$$u=\sin{\left(x \right)}$$$：

$$- \cot{\left(x \right)} + {\color{red}{u}}^{-1} = - \cot{\left(x \right)} + {\color{red}{\sin{\left(x \right)}}}^{-1}$$

因此，

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

加上積分常數：

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

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