$$$\frac{4 \ln\left(2 x\right) - 9}{x}$$$ 的積分

求$$$\int \frac{4 \ln\left(2 x\right) - 9}{x}\, dx$$$。

Expand the expression:

$${\color{red}{\int{\frac{4 \ln{\left(2 x \right)} - 9}{x} d x}}} = {\color{red}{\int{\left(\frac{4 \ln{\left(x \right)}}{x} - \frac{9}{x} + \frac{4 \ln{\left(2 \right)}}{x}\right)d x}}}$$

逐項積分：

$${\color{red}{\int{\left(\frac{4 \ln{\left(x \right)}}{x} - \frac{9}{x} + \frac{4 \ln{\left(2 \right)}}{x}\right)d x}}} = {\color{red}{\left(- \int{\frac{9}{x} d x} + \int{\frac{4 \ln{\left(2 \right)}}{x} d x} + \int{\frac{4 \ln{\left(x \right)}}{x} d x}\right)}}$$

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

$$\int{\frac{4 \ln{\left(2 \right)}}{x} d x} + \int{\frac{4 \ln{\left(x \right)}}{x} d x} - {\color{red}{\int{\frac{9}{x} d x}}} = \int{\frac{4 \ln{\left(2 \right)}}{x} d x} + \int{\frac{4 \ln{\left(x \right)}}{x} d x} - {\color{red}{\left(9 \int{\frac{1}{x} d x}\right)}}$$

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

$$\int{\frac{4 \ln{\left(2 \right)}}{x} d x} + \int{\frac{4 \ln{\left(x \right)}}{x} d x} - 9 {\color{red}{\int{\frac{1}{x} d x}}} = \int{\frac{4 \ln{\left(2 \right)}}{x} d x} + \int{\frac{4 \ln{\left(x \right)}}{x} d x} - 9 {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=4 \ln{\left(2 \right)}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x}$$$：

$$- 9 \ln{\left(\left|{x}\right| \right)} + \int{\frac{4 \ln{\left(x \right)}}{x} d x} + {\color{red}{\int{\frac{4 \ln{\left(2 \right)}}{x} d x}}} = - 9 \ln{\left(\left|{x}\right| \right)} + \int{\frac{4 \ln{\left(x \right)}}{x} d x} + {\color{red}{\left(4 \ln{\left(2 \right)} \int{\frac{1}{x} d x}\right)}}$$

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

$$- 9 \ln{\left(\left|{x}\right| \right)} + \int{\frac{4 \ln{\left(x \right)}}{x} d x} + 4 \ln{\left(2 \right)} {\color{red}{\int{\frac{1}{x} d x}}} = - 9 \ln{\left(\left|{x}\right| \right)} + \int{\frac{4 \ln{\left(x \right)}}{x} d x} + 4 \ln{\left(2 \right)} {\color{red}{\ln{\left(\left|{x}\right| \right)}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=4$$$ 與 $$$f{\left(x \right)} = \frac{\ln{\left(x \right)}}{x}$$$：

$$- 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)} + {\color{red}{\int{\frac{4 \ln{\left(x \right)}}{x} d x}}} = - 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)} + {\color{red}{\left(4 \int{\frac{\ln{\left(x \right)}}{x} d x}\right)}}$$

令 $$$u=\ln{\left(x \right)}$$$。

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

所以，

$$- 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)} + 4 {\color{red}{\int{\frac{\ln{\left(x \right)}}{x} d x}}} = - 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)} + 4 {\color{red}{\int{u d u}}}$$

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

$$- 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)} + 4 {\color{red}{\int{u d u}}}=- 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)} + 4 {\color{red}{\frac{u^{1 + 1}}{1 + 1}}}=- 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)} + 4 {\color{red}{\left(\frac{u^{2}}{2}\right)}}$$

回顧一下 $$$u=\ln{\left(x \right)}$$$：

$$- 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)} + 2 {\color{red}{u}}^{2} = - 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)} + 2 {\color{red}{\ln{\left(x \right)}}}^{2}$$

因此，

$$\int{\frac{4 \ln{\left(2 x \right)} - 9}{x} d x} = 2 \ln{\left(x \right)}^{2} - 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)}$$

加上積分常數：

$$\int{\frac{4 \ln{\left(2 x \right)} - 9}{x} d x} = 2 \ln{\left(x \right)}^{2} - 9 \ln{\left(\left|{x}\right| \right)} + 4 \ln{\left(2 \right)} \ln{\left(\left|{x}\right| \right)}+C$$

$$$\int \frac{4 \ln\left(2 x\right) - 9}{x}\, dx = \left(2 \ln^{2}\left(x\right) - 9 \ln\left(\left|{x}\right|\right) + 4 \ln\left(2\right) \ln\left(\left|{x}\right|\right)\right) + C$$$A