$$$\frac{9}{10 x - 20}$$$ 的積分

求$$$\int \frac{9}{10 x - 20}\, dx$$$。

簡化被積函數:

$${\color{red}{\int{\frac{9}{10 x - 20} d x}}} = {\color{red}{\int{\frac{9}{10 \left(x - 2\right)} d x}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{9}{10}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x - 2}$$$：

$${\color{red}{\int{\frac{9}{10 \left(x - 2\right)} d x}}} = {\color{red}{\left(\frac{9 \int{\frac{1}{x - 2} d x}}{10}\right)}}$$

令 $$$u=x - 2$$$。

則 $$$du=\left(x - 2\right)^{\prime }dx = 1 dx$$$ (步驟見»)，並可得 $$$dx = du$$$。

所以，

$$\frac{9 {\color{red}{\int{\frac{1}{x - 2} d x}}}}{10} = \frac{9 {\color{red}{\int{\frac{1}{u} d u}}}}{10}$$

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

$$\frac{9 {\color{red}{\int{\frac{1}{u} d u}}}}{10} = \frac{9 {\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{10}$$

回顧一下 $$$u=x - 2$$$：

$$\frac{9 \ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{10} = \frac{9 \ln{\left(\left|{{\color{red}{\left(x - 2\right)}}}\right| \right)}}{10}$$

因此，

$$\int{\frac{9}{10 x - 20} d x} = \frac{9 \ln{\left(\left|{x - 2}\right| \right)}}{10}$$

加上積分常數：

$$\int{\frac{9}{10 x - 20} d x} = \frac{9 \ln{\left(\left|{x - 2}\right| \right)}}{10}+C$$

$$$\int \frac{9}{10 x - 20}\, dx = \frac{9 \ln\left(\left|{x - 2}\right|\right)}{10} + C$$$A