$$$- 4 \ln^{2}\left(x\right) + 2 \ln\left(x\right)$$$ 的积分

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

逐项积分：

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

对 $$$c=4$$$ 和 $$$f{\left(x \right)} = \ln{\left(x \right)}^{2}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

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

对于积分$$$\int{\ln{\left(x \right)}^{2} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

设 $$$\operatorname{u}=\ln{\left(x \right)}^{2}$$$ 和 $$$\operatorname{dv}=dx$$$。

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

该积分可以改写为

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

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

$$- 4 x \ln{\left(x \right)}^{2} + 5 {\color{red}{\int{2 \ln{\left(x \right)} d x}}} = - 4 x \ln{\left(x \right)}^{2} + 5 {\color{red}{\left(2 \int{\ln{\left(x \right)} d x}\right)}}$$

对于积分$$$\int{\ln{\left(x \right)} d x}$$$，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 $$$\operatorname{dv}=dx$$$。

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

因此，

$$- 4 x \ln{\left(x \right)}^{2} + 10 {\color{red}{\int{\ln{\left(x \right)} d x}}}=- 4 x \ln{\left(x \right)}^{2} + 10 {\color{red}{\left(\ln{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x} d x}\right)}}=- 4 x \ln{\left(x \right)}^{2} + 10 {\color{red}{\left(x \ln{\left(x \right)} - \int{1 d x}\right)}}$$

应用常数法则 $$$\int c\, dx = c x$$$，使用 $$$c=1$$$：

$$- 4 x \ln{\left(x \right)}^{2} + 10 x \ln{\left(x \right)} - 10 {\color{red}{\int{1 d x}}} = - 4 x \ln{\left(x \right)}^{2} + 10 x \ln{\left(x \right)} - 10 {\color{red}{x}}$$

因此，

$$\int{\left(- 4 \ln{\left(x \right)}^{2} + 2 \ln{\left(x \right)}\right)d x} = - 4 x \ln{\left(x \right)}^{2} + 10 x \ln{\left(x \right)} - 10 x$$

化简：

$$\int{\left(- 4 \ln{\left(x \right)}^{2} + 2 \ln{\left(x \right)}\right)d x} = 2 x \left(- 2 \ln{\left(x \right)}^{2} + 5 \ln{\left(x \right)} - 5\right)$$

加上积分常数：

$$\int{\left(- 4 \ln{\left(x \right)}^{2} + 2 \ln{\left(x \right)}\right)d x} = 2 x \left(- 2 \ln{\left(x \right)}^{2} + 5 \ln{\left(x \right)} - 5\right)+C$$

$$$\int \left(- 4 \ln^{2}\left(x\right) + 2 \ln\left(x\right)\right)\, dx = 2 x \left(- 2 \ln^{2}\left(x\right) + 5 \ln\left(x\right) - 5\right) + C$$$A