$$$\frac{x^{2} - 4}{x}$$$ 的积分

求$$$\int \frac{x^{2} - 4}{x}\, dx$$$。

Expand the expression:

$${\color{red}{\int{\frac{x^{2} - 4}{x} d x}}} = {\color{red}{\int{\left(x - \frac{4}{x}\right)d x}}}$$

逐项积分：

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

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=1$$$：

$$- \int{\frac{4}{x} d x} + {\color{red}{\int{x d x}}}=- \int{\frac{4}{x} d x} + {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \int{\frac{4}{x} d x} + {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

对 $$$c=4$$$ 和 $$$f{\left(x \right)} = \frac{1}{x}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{x^{2}}{2} - {\color{red}{\int{\frac{4}{x} d x}}} = \frac{x^{2}}{2} - {\color{red}{\left(4 \int{\frac{1}{x} d x}\right)}}$$

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

$$\frac{x^{2}}{2} - 4 {\color{red}{\int{\frac{1}{x} d x}}} = \frac{x^{2}}{2} - 4 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

因此，

$$\int{\frac{x^{2} - 4}{x} d x} = \frac{x^{2}}{2} - 4 \ln{\left(\left|{x}\right| \right)}$$

加上积分常数：

$$\int{\frac{x^{2} - 4}{x} d x} = \frac{x^{2}}{2} - 4 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{x^{2} - 4}{x}\, dx = \left(\frac{x^{2}}{2} - 4 \ln\left(\left|{x}\right|\right)\right) + C$$$A