$$$x^{2} + \operatorname{asin}{\left(x \right)}$$$ 的積分

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

逐項積分：

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

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

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

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

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

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

因此，

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

令 $$$u=1 - x^{2}$$$。

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

因此，

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

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

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

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

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

回顧一下 $$$u=1 - x^{2}$$$：

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

因此，

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

加上積分常數：

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

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