$$$\frac{\tan{\left(x \right)}}{23}$$$ 的积分

求$$$\int \frac{\tan{\left(x \right)}}{23}\, dx$$$。

对 $$$c=\frac{1}{23}$$$ 和 $$$f{\left(x \right)} = \tan{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{\frac{\tan{\left(x \right)}}{23} d x}}} = {\color{red}{\left(\frac{\int{\tan{\left(x \right)} d x}}{23}\right)}}$$

将正切表示为 $$$\tan\left(x\right)=\frac{\sin\left(x\right)}{\cos\left(x\right)}$$$:

$$\frac{{\color{red}{\int{\tan{\left(x \right)} d x}}}}{23} = \frac{{\color{red}{\int{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} d x}}}}{23}$$

设$$$u=\cos{\left(x \right)}$$$。

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

该积分可以改写为

$$\frac{{\color{red}{\int{\frac{\sin{\left(x \right)}}{\cos{\left(x \right)}} d x}}}}{23} = \frac{{\color{red}{\int{\left(- \frac{1}{u}\right)d u}}}}{23}$$

对 $$$c=-1$$$ 和 $$$f{\left(u \right)} = \frac{1}{u}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{{\color{red}{\int{\left(- \frac{1}{u}\right)d u}}}}{23} = \frac{{\color{red}{\left(- \int{\frac{1}{u} d u}\right)}}}{23}$$

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

$$- \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{23} = - \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{23}$$

回忆一下 $$$u=\cos{\left(x \right)}$$$:

$$- \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{23} = - \frac{\ln{\left(\left|{{\color{red}{\cos{\left(x \right)}}}}\right| \right)}}{23}$$

因此，

$$\int{\frac{\tan{\left(x \right)}}{23} d x} = - \frac{\ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}}{23}$$

加上积分常数：

$$\int{\frac{\tan{\left(x \right)}}{23} d x} = - \frac{\ln{\left(\left|{\cos{\left(x \right)}}\right| \right)}}{23}+C$$

$$$\int \frac{\tan{\left(x \right)}}{23}\, dx = - \frac{\ln\left(\left|{\cos{\left(x \right)}}\right|\right)}{23} + C$$$A