$$$\frac{x - 9}{x}$$$ 的积分

求$$$\int \frac{x - 9}{x}\, dx$$$。

Expand the expression:

$${\color{red}{\int{\frac{x - 9}{x} d x}}} = {\color{red}{\int{\left(1 - \frac{9}{x}\right)d x}}}$$

逐项积分：

$${\color{red}{\int{\left(1 - \frac{9}{x}\right)d x}}} = {\color{red}{\left(\int{1 d x} - \int{\frac{9}{x} d x}\right)}}$$

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

$$- \int{\frac{9}{x} d x} + {\color{red}{\int{1 d x}}} = - \int{\frac{9}{x} d x} + {\color{red}{x}}$$

对 $$$c=9$$$ 和 $$$f{\left(x \right)} = \frac{1}{x}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$x - {\color{red}{\int{\frac{9}{x} d x}}} = x - {\color{red}{\left(9 \int{\frac{1}{x} d x}\right)}}$$

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

$$x - 9 {\color{red}{\int{\frac{1}{x} d x}}} = x - 9 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

因此，

$$\int{\frac{x - 9}{x} d x} = x - 9 \ln{\left(\left|{x}\right| \right)}$$

加上积分常数：

$$\int{\frac{x - 9}{x} d x} = x - 9 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{x - 9}{x}\, dx = \left(x - 9 \ln\left(\left|{x}\right|\right)\right) + C$$$A