$$$20 x^{2} - \frac{2}{x}$$$ 的積分

求$$$\int \left(20 x^{2} - \frac{2}{x}\right)\, dx$$$。

逐項積分：

$${\color{red}{\int{\left(20 x^{2} - \frac{2}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{2}{x} d x} + \int{20 x^{2} d x}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=2$$$ 與 $$$f{\left(x \right)} = \frac{1}{x}$$$：

$$\int{20 x^{2} d x} - {\color{red}{\int{\frac{2}{x} d x}}} = \int{20 x^{2} d x} - {\color{red}{\left(2 \int{\frac{1}{x} d x}\right)}}$$

$$$\frac{1}{x}$$$ 的積分是 $$$\int{\frac{1}{x} d x} = \ln{\left(\left|{x}\right| \right)}$$$：

$$\int{20 x^{2} d x} - 2 {\color{red}{\int{\frac{1}{x} d x}}} = \int{20 x^{2} d x} - 2 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=20$$$ 與 $$$f{\left(x \right)} = x^{2}$$$：

$$- 2 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\int{20 x^{2} d x}}} = - 2 \ln{\left(\left|{x}\right| \right)} + {\color{red}{\left(20 \int{x^{2} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=2$$$：

$$- 2 \ln{\left(\left|{x}\right| \right)} + 20 {\color{red}{\int{x^{2} d x}}}=- 2 \ln{\left(\left|{x}\right| \right)} + 20 {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=- 2 \ln{\left(\left|{x}\right| \right)} + 20 {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

因此，

$$\int{\left(20 x^{2} - \frac{2}{x}\right)d x} = \frac{20 x^{3}}{3} - 2 \ln{\left(\left|{x}\right| \right)}$$

加上積分常數：

$$\int{\left(20 x^{2} - \frac{2}{x}\right)d x} = \frac{20 x^{3}}{3} - 2 \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \left(20 x^{2} - \frac{2}{x}\right)\, dx = \left(\frac{20 x^{3}}{3} - 2 \ln\left(\left|{x}\right|\right)\right) + C$$$A