$$$x^{2} \sin{\left(x \right)}$$$ 的積分

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

對於積分 $$$\int{x^{2} \sin{\left(x \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

令 $$$\operatorname{u}=x^{2}$$$ 與 $$$\operatorname{dv}=\sin{\left(x \right)} dx$$$。

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

因此，

$${\color{red}{\int{x^{2} \sin{\left(x \right)} d x}}}={\color{red}{\left(x^{2} \cdot \left(- \cos{\left(x \right)}\right)-\int{\left(- \cos{\left(x \right)}\right) \cdot 2 x d x}\right)}}={\color{red}{\left(- x^{2} \cos{\left(x \right)} - \int{\left(- 2 x \cos{\left(x \right)}\right)d x}\right)}}$$

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

$$- x^{2} \cos{\left(x \right)} - {\color{red}{\int{\left(- 2 x \cos{\left(x \right)}\right)d x}}} = - x^{2} \cos{\left(x \right)} - {\color{red}{\left(- 2 \int{x \cos{\left(x \right)} d x}\right)}}$$

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

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

則 $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{\cos{\left(x \right)} d x}=\sin{\left(x \right)}$$$（步驟見 »）。

該積分可改寫為

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

正弦函數的積分為 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$：

$$- x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} - 2 {\color{red}{\int{\sin{\left(x \right)} d x}}} = - x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} - 2 {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$

因此，

$$\int{x^{2} \sin{\left(x \right)} d x} = - x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + 2 \cos{\left(x \right)}$$

加上積分常數：

$$\int{x^{2} \sin{\left(x \right)} d x} = - x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + 2 \cos{\left(x \right)}+C$$

$$$\int x^{2} \sin{\left(x \right)}\, dx = \left(- x^{2} \cos{\left(x \right)} + 2 x \sin{\left(x \right)} + 2 \cos{\left(x \right)}\right) + C$$$A