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

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

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

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

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 $$$\frac{dx}{x} = du$$$。

該積分可改寫為

$$\frac{{\color{red}{\int{\frac{\ln{\left(x \right)}}{x} d x}}}}{2} = \frac{{\color{red}{\int{u d u}}}}{2}$$

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

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

回顧一下 $$$u=\ln{\left(x \right)}$$$：

$$\frac{{\color{red}{u}}^{2}}{4} = \frac{{\color{red}{\ln{\left(x \right)}}}^{2}}{4}$$

因此，

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

加上積分常數：

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

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