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

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

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

$${\color{red}{\int{2 \operatorname{atan}{\left(x \right)} d x}}} = {\color{red}{\left(2 \int{\operatorname{atan}{\left(x \right)} d x}\right)}}$$

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

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

该积分可以改写为

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

设$$$u=x^{2} + 1$$$。

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

该积分可以改写为

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

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

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

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

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

回忆一下 $$$u=x^{2} + 1$$$:

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

因此，

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

加上积分常数：

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

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