$$$- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1$$$ 的積分

求$$$\int \left(- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1\right)\, dx$$$。

逐項積分：

$${\color{red}{\int{\left(- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1\right)d x}}} = {\color{red}{\left(- \int{1 d x} - \int{2 \csc^{3}{\left(x \right)} d x} + \int{\sec^{2}{\left(x \right)} d x}\right)}}$$

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

$$- \int{2 \csc^{3}{\left(x \right)} d x} + \int{\sec^{2}{\left(x \right)} d x} - {\color{red}{\int{1 d x}}} = - \int{2 \csc^{3}{\left(x \right)} d x} + \int{\sec^{2}{\left(x \right)} d x} - {\color{red}{x}}$$

$$$\sec^{2}{\left(x \right)}$$$ 的積分是 $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$：

$$- x - \int{2 \csc^{3}{\left(x \right)} d x} + {\color{red}{\int{\sec^{2}{\left(x \right)} d x}}} = - x - \int{2 \csc^{3}{\left(x \right)} d x} + {\color{red}{\tan{\left(x \right)}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=2$$$ 與 $$$f{\left(x \right)} = \csc^{3}{\left(x \right)}$$$：

$$- x + \tan{\left(x \right)} - {\color{red}{\int{2 \csc^{3}{\left(x \right)} d x}}} = - x + \tan{\left(x \right)} - {\color{red}{\left(2 \int{\csc^{3}{\left(x \right)} d x}\right)}}$$

對於積分 $$$\int{\csc^{3}{\left(x \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

令 $$$\operatorname{u}=\csc{\left(x \right)}$$$ 與 $$$\operatorname{dv}=\csc^{2}{\left(x \right)} dx$$$。

則 $$$\operatorname{du}=\left(\csc{\left(x \right)}\right)^{\prime }dx=- \cot{\left(x \right)} \csc{\left(x \right)} dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{\csc^{2}{\left(x \right)} d x}=- \cot{\left(x \right)}$$$（步驟見 »）。

所以，

$$\int{\csc^{3}{\left(x \right)} d x}=\csc{\left(x \right)} \cdot \left(- \cot{\left(x \right)}\right)-\int{\left(- \cot{\left(x \right)}\right) \cdot \left(- \cot{\left(x \right)} \csc{\left(x \right)}\right) d x}=- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\cot^{2}{\left(x \right)} \csc{\left(x \right)} d x}$$

套用公式 $$$\cot^{2}{\left(x \right)} = \csc^{2}{\left(x \right)} - 1$$$:

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

展開：

$$- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{2}{\left(x \right)} - 1\right) \csc{\left(x \right)} d x}=- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{3}{\left(x \right)} - \csc{\left(x \right)}\right)d x}$$

差的積分等於積分之差：

$$- \cot{\left(x \right)} \csc{\left(x \right)} - \int{\left(\csc^{3}{\left(x \right)} - \csc{\left(x \right)}\right)d x}=- \cot{\left(x \right)} \csc{\left(x \right)} + \int{\csc{\left(x \right)} d x} - \int{\csc^{3}{\left(x \right)} d x}$$

因此，我們得到關於該積分的下列簡單線性方程：

$${\color{red}{\int{\csc^{3}{\left(x \right)} d x}}}=- \cot{\left(x \right)} \csc{\left(x \right)} + \int{\csc{\left(x \right)} d x} - {\color{red}{\int{\csc^{3}{\left(x \right)} d x}}}$$

解得

$$\int{\csc^{3}{\left(x \right)} d x}=- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{\int{\csc{\left(x \right)} d x}}{2}$$

因此，

$$- x + \tan{\left(x \right)} - 2 {\color{red}{\int{\csc^{3}{\left(x \right)} d x}}} = - x + \tan{\left(x \right)} - 2 {\color{red}{\left(- \frac{\cot{\left(x \right)} \csc{\left(x \right)}}{2} + \frac{\int{\csc{\left(x \right)} d x}}{2}\right)}}$$

將餘割改寫為 $$$\csc\left(x\right)=\frac{1}{\sin\left(x\right)}$$$:

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

使用倍角公式 $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$ 重寫正弦:

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

將分子與分母同時乘以 $$$\sec^2\left(\frac{x}{2} \right)$$$:

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

令 $$$u=\tan{\left(\frac{x}{2} \right)}$$$。

則 $$$du=\left(\tan{\left(\frac{x}{2} \right)}\right)^{\prime }dx = \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2} dx$$$ (步驟見»)，並可得 $$$\sec^{2}{\left(\frac{x}{2} \right)} dx = 2 du$$$。

因此，

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

$$$\frac{1}{u}$$$ 的積分是 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$：

$$- x + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)} - {\color{red}{\int{\frac{1}{u} d u}}} = - x + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)} - {\color{red}{\ln{\left(\left|{u}\right| \right)}}}$$

回顧一下 $$$u=\tan{\left(\frac{x}{2} \right)}$$$：

$$- x - \ln{\left(\left|{{\color{red}{u}}}\right| \right)} + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)} = - x - \ln{\left(\left|{{\color{red}{\tan{\left(\frac{x}{2} \right)}}}}\right| \right)} + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)}$$

因此，

$$\int{\left(- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1\right)d x} = - x - \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)}$$

加上積分常數：

$$\int{\left(- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1\right)d x} = - x - \ln{\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right| \right)} + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)}+C$$

$$$\int \left(- 2 \csc^{3}{\left(x \right)} + \sec^{2}{\left(x \right)} - 1\right)\, dx = \left(- x - \ln\left(\left|{\tan{\left(\frac{x}{2} \right)}}\right|\right) + \tan{\left(x \right)} + \cot{\left(x \right)} \csc{\left(x \right)}\right) + C$$$A