$$$x^{2} \operatorname{atan}{\left(x \right)}$$$ 的积分

求$$$\int x^{2} \operatorname{atan}{\left(x \right)}\, dx$$$。

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

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

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

所以，

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

化简被积函数:

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

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

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

由于分子次数不小于分母次数，进行多项式长除法（步骤见»）:

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

逐项积分：

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

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

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

设$$$u=x^{2} + 1$$$。

则$$$du=\left(x^{2} + 1\right)^{\prime }dx = 2 x dx$$$ (步骤见»)，并有$$$x dx = \frac{du}{2}$$$。

所以，

$$\frac{x^{3} \operatorname{atan}{\left(x \right)}}{3} - \frac{x^{2}}{6} + \frac{{\color{red}{\int{\frac{x}{x^{2} + 1} d x}}}}{3} = \frac{x^{3} \operatorname{atan}{\left(x \right)}}{3} - \frac{x^{2}}{6} + \frac{{\color{red}{\int{\frac{1}{2 u} d u}}}}{3}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(u \right)} = \frac{1}{u}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{x^{3} \operatorname{atan}{\left(x \right)}}{3} - \frac{x^{2}}{6} + \frac{{\color{red}{\int{\frac{1}{2 u} d u}}}}{3} = \frac{x^{3} \operatorname{atan}{\left(x \right)}}{3} - \frac{x^{2}}{6} + \frac{{\color{red}{\left(\frac{\int{\frac{1}{u} d u}}{2}\right)}}}{3}$$

$$$\frac{1}{u}$$$ 的积分为 $$$\int{\frac{1}{u} d u} = \ln{\left(\left|{u}\right| \right)}$$$:

$$\frac{x^{3} \operatorname{atan}{\left(x \right)}}{3} - \frac{x^{2}}{6} + \frac{{\color{red}{\int{\frac{1}{u} d u}}}}{6} = \frac{x^{3} \operatorname{atan}{\left(x \right)}}{3} - \frac{x^{2}}{6} + \frac{{\color{red}{\ln{\left(\left|{u}\right| \right)}}}}{6}$$

回忆一下 $$$u=x^{2} + 1$$$:

$$\frac{x^{3} \operatorname{atan}{\left(x \right)}}{3} - \frac{x^{2}}{6} + \frac{\ln{\left(\left|{{\color{red}{u}}}\right| \right)}}{6} = \frac{x^{3} \operatorname{atan}{\left(x \right)}}{3} - \frac{x^{2}}{6} + \frac{\ln{\left(\left|{{\color{red}{\left(x^{2} + 1\right)}}}\right| \right)}}{6}$$

因此，

$$\int{x^{2} \operatorname{atan}{\left(x \right)} d x} = \frac{x^{3} \operatorname{atan}{\left(x \right)}}{3} - \frac{x^{2}}{6} + \frac{\ln{\left(x^{2} + 1 \right)}}{6}$$

化简：

$$\int{x^{2} \operatorname{atan}{\left(x \right)} d x} = \frac{2 x^{3} \operatorname{atan}{\left(x \right)} - x^{2} + \ln{\left(x^{2} + 1 \right)}}{6}$$

加上积分常数：

$$\int{x^{2} \operatorname{atan}{\left(x \right)} d x} = \frac{2 x^{3} \operatorname{atan}{\left(x \right)} - x^{2} + \ln{\left(x^{2} + 1 \right)}}{6}+C$$

$$$\int x^{2} \operatorname{atan}{\left(x \right)}\, dx = \frac{2 x^{3} \operatorname{atan}{\left(x \right)} - x^{2} + \ln\left(x^{2} + 1\right)}{6} + C$$$A