$$$\frac{\ln\left(x\right)}{x^{3}}$$$ 的积分

求$$$\int \frac{\ln\left(x\right)}{x^{3}}\, dx$$$。

对于积分$$$\int{\frac{\ln{\left(x \right)}}{x^{3}} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

设 $$$\operatorname{u}=\ln{\left(x \right)}$$$ 和 $$$\operatorname{dv}=\frac{dx}{x^{3}}$$$。

则 $$$\operatorname{du}=\left(\ln{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x}$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{\frac{1}{x^{3}} d x}=- \frac{1}{2 x^{2}}$$$ (步骤见 »)。

因此，

$${\color{red}{\int{\frac{\ln{\left(x \right)}}{x^{3}} d x}}}={\color{red}{\left(\ln{\left(x \right)} \cdot \left(- \frac{1}{2 x^{2}}\right)-\int{\left(- \frac{1}{2 x^{2}}\right) \cdot \frac{1}{x} d x}\right)}}={\color{red}{\left(- \int{\left(- \frac{1}{2 x^{3}}\right)d x} - \frac{\ln{\left(x \right)}}{2 x^{2}}\right)}}$$

对 $$$c=- \frac{1}{2}$$$ 和 $$$f{\left(x \right)} = \frac{1}{x^{3}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$- {\color{red}{\int{\left(- \frac{1}{2 x^{3}}\right)d x}}} - \frac{\ln{\left(x \right)}}{2 x^{2}} = - {\color{red}{\left(- \frac{\int{\frac{1}{x^{3}} d x}}{2}\right)}} - \frac{\ln{\left(x \right)}}{2 x^{2}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-3$$$：

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

因此，

$$\int{\frac{\ln{\left(x \right)}}{x^{3}} d x} = - \frac{\ln{\left(x \right)}}{2 x^{2}} - \frac{1}{4 x^{2}}$$

化简：

$$\int{\frac{\ln{\left(x \right)}}{x^{3}} d x} = \frac{- 2 \ln{\left(x \right)} - 1}{4 x^{2}}$$

加上积分常数：

$$\int{\frac{\ln{\left(x \right)}}{x^{3}} d x} = \frac{- 2 \ln{\left(x \right)} - 1}{4 x^{2}}+C$$

$$$\int \frac{\ln\left(x\right)}{x^{3}}\, dx = \frac{- 2 \ln\left(x\right) - 1}{4 x^{2}} + C$$$A