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

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

逐项积分：

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

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=1$$$：

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

$$$\sec^{2}{\left(x \right)}$$$ 的积分为 $$$\int{\sec^{2}{\left(x \right)} d x} = \tan{\left(x \right)}$$$:

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

对 $$$c=2$$$ 和 $$$f{\left(x \right)} = \csc^{3}{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

对于积分$$$\int{\csc^{3}{\left(x \right)} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

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

则 $$$\operatorname{du}=\left(\csc{\left(x \right)}\right)^{\prime }dx=- \cot{\left(x \right)} \csc{\left(x \right)} dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{\csc^{2}{\left(x \right)} d x}=- \cot{\left(x \right)}$$$ (步骤见 »)。

该积分可以改写为

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

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

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

展开：

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

差的积分等于积分的差：

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

因此，我们得到关于该积分的以下简单线性方程：

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

求解可得

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

因此，

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

将余割改写为$$$\csc\left(x\right)=\frac{1}{\sin\left(x\right)}$$$:

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

使用二倍角公式 $$$\sin\left(x\right)=2\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)$$$ 改写正弦:

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

将分子和分母同时乘以 $$$\sec^2\left(\frac{x}{2} \right)$$$:

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

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

则$$$du=\left(\tan{\left(\frac{x}{2} \right)}\right)^{\prime }dx = \frac{\sec^{2}{\left(\frac{x}{2} \right)}}{2} dx$$$ (步骤见»)，并有$$$\sec^{2}{\left(\frac{x}{2} \right)} dx = 2 du$$$。

所以，

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

$$$\frac{1}{u}$$$ 的积分为 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

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

回忆一下 $$$u=\tan{\left(\frac{x}{2} \right)}$$$:

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

因此，

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

加上积分常数：

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

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