$$$\cot{\left(x + \frac{\pi}{4} \right)}$$$ 的积分

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

设$$$u=x + \frac{\pi}{4}$$$。

则$$$du=\left(x + \frac{\pi}{4}\right)^{\prime }dx = 1 dx$$$ (步骤见»)，并有$$$dx = du$$$。

积分变为

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

将余切改写为 $$$\cot\left( u \right)=\frac{\cos\left( u \right)}{\sin\left( u \right)}$$$:

$${\color{red}{\int{\cot{\left(u \right)} d u}}} = {\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}}$$

设$$$v=\sin{\left(u \right)}$$$。

则$$$dv=\left(\sin{\left(u \right)}\right)^{\prime }du = \cos{\left(u \right)} du$$$ (步骤见»)，并有$$$\cos{\left(u \right)} du = dv$$$。

因此，

$${\color{red}{\int{\frac{\cos{\left(u \right)}}{\sin{\left(u \right)}} d u}}} = {\color{red}{\int{\frac{1}{v} d v}}}$$

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

$${\color{red}{\int{\frac{1}{v} d v}}} = {\color{red}{\ln{\left(\left|{v}\right| \right)}}}$$

回忆一下 $$$v=\sin{\left(u \right)}$$$:

$$\ln{\left(\left|{{\color{red}{v}}}\right| \right)} = \ln{\left(\left|{{\color{red}{\sin{\left(u \right)}}}}\right| \right)}$$

回忆一下 $$$u=x + \frac{\pi}{4}$$$:

$$\ln{\left(\left|{\sin{\left({\color{red}{u}} \right)}}\right| \right)} = \ln{\left(\left|{\sin{\left({\color{red}{\left(x + \frac{\pi}{4}\right)}} \right)}}\right| \right)}$$

因此，

$$\int{\cot{\left(x + \frac{\pi}{4} \right)} d x} = \ln{\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right| \right)}$$

加上积分常数：

$$\int{\cot{\left(x + \frac{\pi}{4} \right)} d x} = \ln{\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right| \right)}+C$$

$$$\int \cot{\left(x + \frac{\pi}{4} \right)}\, dx = \ln\left(\left|{\sin{\left(x + \frac{\pi}{4} \right)}}\right|\right) + C$$$A