$$$\operatorname{atan}{\left(\sqrt{x} \right)}$$$ 的積分

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

對於積分 $$$\int{\operatorname{atan}{\left(\sqrt{x} \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

令 $$$\operatorname{u}=\operatorname{atan}{\left(\sqrt{x} \right)}$$$ 與 $$$\operatorname{dv}=dx$$$。

則 $$$\operatorname{du}=\left(\operatorname{atan}{\left(\sqrt{x} \right)}\right)^{\prime }dx=\frac{1}{2 \sqrt{x} \left(x + 1\right)} dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{1 d x}=x$$$（步驟見 »）。

因此，

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

簡化被積函數:

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

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{1}{2}$$$ 與 $$$f{\left(x \right)} = \frac{\sqrt{x}}{x + 1}$$$：

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

令 $$$u=\sqrt{x}$$$。

則 $$$du=\left(\sqrt{x}\right)^{\prime }dx = \frac{1}{2 \sqrt{x}} dx$$$ (步驟見»)，並可得 $$$\frac{dx}{\sqrt{x}} = 2 du$$$。

因此，

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

套用常數倍法則 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$，使用 $$$c=2$$$ 與 $$$f{\left(u \right)} = \frac{u^{2}}{u^{2} + 1}$$$：

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

重寫並拆分分式:

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

逐項積分：

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

配合 $$$c=1$$$，應用常數法則 $$$\int c\, du = c u$$$：

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

$$$\frac{1}{u^{2} + 1}$$$ 的積分是 $$$\int{\frac{1}{u^{2} + 1} d u} = \operatorname{atan}{\left(u \right)}$$$：

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

回顧一下 $$$u=\sqrt{x}$$$：

$$x \operatorname{atan}{\left(\sqrt{x} \right)} + \operatorname{atan}{\left({\color{red}{u}} \right)} - {\color{red}{u}} = x \operatorname{atan}{\left(\sqrt{x} \right)} + \operatorname{atan}{\left({\color{red}{\sqrt{x}}} \right)} - {\color{red}{\sqrt{x}}}$$

因此，

$$\int{\operatorname{atan}{\left(\sqrt{x} \right)} d x} = - \sqrt{x} + x \operatorname{atan}{\left(\sqrt{x} \right)} + \operatorname{atan}{\left(\sqrt{x} \right)}$$

加上積分常數：

$$\int{\operatorname{atan}{\left(\sqrt{x} \right)} d x} = - \sqrt{x} + x \operatorname{atan}{\left(\sqrt{x} \right)} + \operatorname{atan}{\left(\sqrt{x} \right)}+C$$

$$$\int \operatorname{atan}{\left(\sqrt{x} \right)}\, dx = \left(- \sqrt{x} + x \operatorname{atan}{\left(\sqrt{x} \right)} + \operatorname{atan}{\left(\sqrt{x} \right)}\right) + C$$$A