$$$\cot^{2}{\left(x + \frac{\pi}{4} \right)}$$$ 的積分

求$$$\int \cot^{2}{\left(x + \frac{\pi}{4} \right)}\, dx$$$。

令 $$$u=x + \frac{\pi}{4}$$$。

則 $$$du=\left(x + \frac{\pi}{4}\right)^{\prime }dx = 1 dx$$$ (步驟見»)，並可得 $$$dx = du$$$。

該積分可改寫為

$${\color{red}{\int{\cot^{2}{\left(x + \frac{\pi}{4} \right)} d x}}} = {\color{red}{\int{\cot^{2}{\left(u \right)} d u}}}$$

令 $$$v=\cot{\left(u \right)}$$$。

則 $$$dv=\left(\cot{\left(u \right)}\right)^{\prime }du = - \csc^{2}{\left(u \right)} du$$$ (步驟見»)，並可得 $$$\csc^{2}{\left(u \right)} du = - dv$$$。

因此，

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

套用常數倍法則 $$$\int c f{\left(v \right)}\, dv = c \int f{\left(v \right)}\, dv$$$，使用 $$$c=-1$$$ 與 $$$f{\left(v \right)} = \frac{v^{2}}{v^{2} + 1}$$$：

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

重寫並拆分分式:

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

逐項積分：

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

配合 $$$c=1$$$，應用常數法則 $$$\int c\, dv = c v$$$：

$$\int{\frac{1}{v^{2} + 1} d v} - {\color{red}{\int{1 d v}}} = \int{\frac{1}{v^{2} + 1} d v} - {\color{red}{v}}$$

$$$\frac{1}{v^{2} + 1}$$$ 的積分是 $$$\int{\frac{1}{v^{2} + 1} d v} = \operatorname{atan}{\left(v \right)}$$$：

$$- v + {\color{red}{\int{\frac{1}{v^{2} + 1} d v}}} = - v + {\color{red}{\operatorname{atan}{\left(v \right)}}}$$

回顧一下 $$$v=\cot{\left(u \right)}$$$：

$$\operatorname{atan}{\left({\color{red}{v}} \right)} - {\color{red}{v}} = \operatorname{atan}{\left({\color{red}{\cot{\left(u \right)}}} \right)} - {\color{red}{\cot{\left(u \right)}}}$$

回顧一下 $$$u=x + \frac{\pi}{4}$$$：

$$- \cot{\left({\color{red}{u}} \right)} + \operatorname{atan}{\left(\cot{\left({\color{red}{u}} \right)} \right)} = - \cot{\left({\color{red}{\left(x + \frac{\pi}{4}\right)}} \right)} + \operatorname{atan}{\left(\cot{\left({\color{red}{\left(x + \frac{\pi}{4}\right)}} \right)} \right)}$$

因此，

$$\int{\cot^{2}{\left(x + \frac{\pi}{4} \right)} d x} = - \cot{\left(x + \frac{\pi}{4} \right)} + \operatorname{atan}{\left(\cot{\left(x + \frac{\pi}{4} \right)} \right)}$$

加上積分常數：

$$\int{\cot^{2}{\left(x + \frac{\pi}{4} \right)} d x} = - \cot{\left(x + \frac{\pi}{4} \right)} + \operatorname{atan}{\left(\cot{\left(x + \frac{\pi}{4} \right)} \right)}+C$$

$$$\int \cot^{2}{\left(x + \frac{\pi}{4} \right)}\, dx = \left(- \cot{\left(x + \frac{\pi}{4} \right)} + \operatorname{atan}{\left(\cot{\left(x + \frac{\pi}{4} \right)} \right)}\right) + C$$$A