$$$x^{2} \operatorname{atan}{\left(\sqrt{x} \right)}$$$ 的積分

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

對於積分 $$$\int{x^{2} \operatorname{atan}{\left(\sqrt{x} \right)} d x}$$$，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 $$$\operatorname{dv}=x^{2} dx$$$。

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 $$$\operatorname{v}=\int{x^{2} d x}=\frac{x^{3}}{3}$$$（步驟見 »）。

該積分變為

$${\color{red}{\int{x^{2} \operatorname{atan}{\left(\sqrt{x} \right)} d x}}}={\color{red}{\left(\operatorname{atan}{\left(\sqrt{x} \right)} \cdot \frac{x^{3}}{3}-\int{\frac{x^{3}}{3} \cdot \frac{1}{2 \sqrt{x} \left(x + 1\right)} d x}\right)}}={\color{red}{\left(\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \int{\frac{x^{\frac{5}{2}}}{6 x + 6} d x}\right)}}$$

簡化被積函數:

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - {\color{red}{\int{\frac{x^{\frac{5}{2}}}{6 x + 6} d x}}} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - {\color{red}{\int{\frac{x^{\frac{5}{2}}}{6 \left(x + 1\right)} d x}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{1}{6}$$$ 與 $$$f{\left(x \right)} = \frac{x^{\frac{5}{2}}}{x + 1}$$$：

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - {\color{red}{\int{\frac{x^{\frac{5}{2}}}{6 \left(x + 1\right)} d x}}} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - {\color{red}{\left(\frac{\int{\frac{x^{\frac{5}{2}}}{x + 1} d x}}{6}\right)}}$$

令 $$$u=\sqrt{x}$$$。

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

該積分變為

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\int{\frac{x^{\frac{5}{2}}}{x + 1} d x}}}}{6} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\int{\frac{2 u^{6}}{u^{2} + 1} d u}}}}{6}$$

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 $$$f{\left(u \right)} = \frac{u^{6}}{u^{2} + 1}$$$：

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

由於分子次數不小於分母次數，進行多項式長除法（步驟見»）:

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\int{\frac{u^{6}}{u^{2} + 1} d u}}}}{3} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\int{\left(u^{4} - u^{2} + 1 - \frac{1}{u^{2} + 1}\right)d u}}}}{3}$$

逐項積分：

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\int{\left(u^{4} - u^{2} + 1 - \frac{1}{u^{2} + 1}\right)d u}}}}{3} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} - \frac{{\color{red}{\left(\int{1 d u} - \int{u^{2} d u} + \int{u^{4} d u} - \int{\frac{1}{u^{2} + 1} d u}\right)}}}{3}$$

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

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{u^{2} d u}}{3} - \frac{\int{u^{4} d u}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} - \frac{{\color{red}{\int{1 d u}}}}{3} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{u^{2} d u}}{3} - \frac{\int{u^{4} d u}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} - \frac{{\color{red}{u}}}{3}$$

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=4$$$：

$$- \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{u^{2} d u}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} - \frac{{\color{red}{\int{u^{4} d u}}}}{3}=- \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{u^{2} d u}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} - \frac{{\color{red}{\frac{u^{1 + 4}}{1 + 4}}}}{3}=- \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{u^{2} d u}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} - \frac{{\color{red}{\left(\frac{u^{5}}{5}\right)}}}{3}$$

套用冪次法則 $$$\int u^{n}\, du = \frac{u^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=2$$$：

$$- \frac{u^{5}}{15} - \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} + \frac{{\color{red}{\int{u^{2} d u}}}}{3}=- \frac{u^{5}}{15} - \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} + \frac{{\color{red}{\frac{u^{1 + 2}}{1 + 2}}}}{3}=- \frac{u^{5}}{15} - \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\int{\frac{1}{u^{2} + 1} d u}}{3} + \frac{{\color{red}{\left(\frac{u^{3}}{3}\right)}}}{3}$$

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

$$- \frac{u^{5}}{15} + \frac{u^{3}}{9} - \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{{\color{red}{\int{\frac{1}{u^{2} + 1} d u}}}}{3} = - \frac{u^{5}}{15} + \frac{u^{3}}{9} - \frac{u}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{{\color{red}{\operatorname{atan}{\left(u \right)}}}}{3}$$

回顧一下 $$$u=\sqrt{x}$$$：

$$\frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\operatorname{atan}{\left({\color{red}{u}} \right)}}{3} - \frac{{\color{red}{u}}}{3} + \frac{{\color{red}{u}}^{3}}{9} - \frac{{\color{red}{u}}^{5}}{15} = \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\operatorname{atan}{\left({\color{red}{\sqrt{x}}} \right)}}{3} - \frac{{\color{red}{\sqrt{x}}}}{3} + \frac{{\color{red}{\sqrt{x}}}^{3}}{9} - \frac{{\color{red}{\sqrt{x}}}^{5}}{15}$$

因此，

$$\int{x^{2} \operatorname{atan}{\left(\sqrt{x} \right)} d x} = - \frac{x^{\frac{5}{2}}}{15} + \frac{x^{\frac{3}{2}}}{9} - \frac{\sqrt{x}}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{3}$$

加上積分常數：

$$\int{x^{2} \operatorname{atan}{\left(\sqrt{x} \right)} d x} = - \frac{x^{\frac{5}{2}}}{15} + \frac{x^{\frac{3}{2}}}{9} - \frac{\sqrt{x}}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{3}+C$$

$$$\int x^{2} \operatorname{atan}{\left(\sqrt{x} \right)}\, dx = \left(- \frac{x^{\frac{5}{2}}}{15} + \frac{x^{\frac{3}{2}}}{9} - \frac{\sqrt{x}}{3} + \frac{x^{3} \operatorname{atan}{\left(\sqrt{x} \right)}}{3} + \frac{\operatorname{atan}{\left(\sqrt{x} \right)}}{3}\right) + C$$$A