$$$\frac{\ln^{2}\left(t\right)}{t}$$$ 的积分

求$$$\int \frac{\ln^{2}\left(t\right)}{t}\, dt$$$。

设$$$u=\ln{\left(t \right)}$$$。

则$$$du=\left(\ln{\left(t \right)}\right)^{\prime }dt = \frac{dt}{t}$$$ (步骤见»)，并有$$$\frac{dt}{t} = du$$$。

该积分可以改写为

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

应用幂法则 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=2$$$：

$${\color{red}{\int{u^{2} d u}}}={\color{red}{\frac{u^{1 + 2}}{1 + 2}}}={\color{red}{\left(\frac{u^{3}}{3}\right)}}$$

回忆一下 $$$u=\ln{\left(t \right)}$$$:

$$\frac{{\color{red}{u}}^{3}}{3} = \frac{{\color{red}{\ln{\left(t \right)}}}^{3}}{3}$$

因此，

$$\int{\frac{\ln{\left(t \right)}^{2}}{t} d t} = \frac{\ln{\left(t \right)}^{3}}{3}$$

加上积分常数：

$$\int{\frac{\ln{\left(t \right)}^{2}}{t} d t} = \frac{\ln{\left(t \right)}^{3}}{3}+C$$

$$$\int \frac{\ln^{2}\left(t\right)}{t}\, dt = \frac{\ln^{3}\left(t\right)}{3} + C$$$A