$$$\frac{\ln^{12}\left(x\right)}{x}$$$ 的積分

求$$$\int \frac{\ln^{12}\left(x\right)}{x}\, dx$$$。

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 $$$\frac{dx}{x} = du$$$。

該積分變為

$${\color{red}{\int{\frac{\ln{\left(x \right)}^{12}}{x} d x}}} = {\color{red}{\int{u^{12} d u}}}$$

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=12$$$：

$${\color{red}{\int{u^{12} d u}}}={\color{red}{\frac{u^{1 + 12}}{1 + 12}}}={\color{red}{\left(\frac{u^{13}}{13}\right)}}$$

回顧一下 $$$u=\ln{\left(x \right)}$$$：

$$\frac{{\color{red}{u}}^{13}}{13} = \frac{{\color{red}{\ln{\left(x \right)}}}^{13}}{13}$$

因此，

$$\int{\frac{\ln{\left(x \right)}^{12}}{x} d x} = \frac{\ln{\left(x \right)}^{13}}{13}$$

加上積分常數：

$$\int{\frac{\ln{\left(x \right)}^{12}}{x} d x} = \frac{\ln{\left(x \right)}^{13}}{13}+C$$

$$$\int \frac{\ln^{12}\left(x\right)}{x}\, dx = \frac{\ln^{13}\left(x\right)}{13} + C$$$A