$$$x^{2} \sin{\left(x \right)}$$$ 的积分

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

对于积分$$$\int{x^{2} \sin{\left(x \right)} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

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

则 $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{\sin{\left(x \right)} d x}=- \cos{\left(x \right)}$$$ (步骤见 »)。

积分变为

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

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

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

对于积分$$$\int{x \cos{\left(x \right)} d x}$$$，使用分部积分法$$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

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

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

积分变为

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

正弦函数的积分为 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:

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

因此，

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

加上积分常数：

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

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