$$$x^{2} \cos{\left(3 x \right)}$$$ 的积分

求$$$\int x^{2} \cos{\left(3 x \right)}\, dx$$$。

对于积分$$$\int{x^{2} \cos{\left(3 x \right)} d x}$$$，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 $$$\operatorname{dv}=\cos{\left(3 x \right)} dx$$$。

则 $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{\cos{\left(3 x \right)} d x}=\frac{\sin{\left(3 x \right)}}{3}$$$ (步骤见 »)。

所以，

$${\color{red}{\int{x^{2} \cos{\left(3 x \right)} d x}}}={\color{red}{\left(x^{2} \cdot \frac{\sin{\left(3 x \right)}}{3}-\int{\frac{\sin{\left(3 x \right)}}{3} \cdot 2 x d x}\right)}}={\color{red}{\left(\frac{x^{2} \sin{\left(3 x \right)}}{3} - \int{\frac{2 x \sin{\left(3 x \right)}}{3} d x}\right)}}$$

对 $$$c=\frac{2}{3}$$$ 和 $$$f{\left(x \right)} = x \sin{\left(3 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} - {\color{red}{\int{\frac{2 x \sin{\left(3 x \right)}}{3} d x}}} = \frac{x^{2} \sin{\left(3 x \right)}}{3} - {\color{red}{\left(\frac{2 \int{x \sin{\left(3 x \right)} d x}}{3}\right)}}$$

对于积分$$$\int{x \sin{\left(3 x \right)} d x}$$$，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 $$$\operatorname{dv}=\sin{\left(3 x \right)} dx$$$。

则 $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{\sin{\left(3 x \right)} d x}=- \frac{\cos{\left(3 x \right)}}{3}$$$ (步骤见 »)。

所以，

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} - \frac{2 {\color{red}{\int{x \sin{\left(3 x \right)} d x}}}}{3}=\frac{x^{2} \sin{\left(3 x \right)}}{3} - \frac{2 {\color{red}{\left(x \cdot \left(- \frac{\cos{\left(3 x \right)}}{3}\right)-\int{\left(- \frac{\cos{\left(3 x \right)}}{3}\right) \cdot 1 d x}\right)}}}{3}=\frac{x^{2} \sin{\left(3 x \right)}}{3} - \frac{2 {\color{red}{\left(- \frac{x \cos{\left(3 x \right)}}{3} - \int{\left(- \frac{\cos{\left(3 x \right)}}{3}\right)d x}\right)}}}{3}$$

对 $$$c=- \frac{1}{3}$$$ 和 $$$f{\left(x \right)} = \cos{\left(3 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} + \frac{2 {\color{red}{\int{\left(- \frac{\cos{\left(3 x \right)}}{3}\right)d x}}}}{3} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} + \frac{2 {\color{red}{\left(- \frac{\int{\cos{\left(3 x \right)} d x}}{3}\right)}}}{3}$$

设$$$u=3 x$$$。

则$$$du=\left(3 x\right)^{\prime }dx = 3 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{3}$$$。

因此，

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\int{\cos{\left(3 x \right)} d x}}}}{9} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\int{\frac{\cos{\left(u \right)}}{3} d u}}}}{9}$$

对 $$$c=\frac{1}{3}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\int{\frac{\cos{\left(u \right)}}{3} d u}}}}{9} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{3}\right)}}}{9}$$

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

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\int{\cos{\left(u \right)} d u}}}}{27} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 {\color{red}{\sin{\left(u \right)}}}}{27}$$

回忆一下 $$$u=3 x$$$:

$$\frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 \sin{\left({\color{red}{u}} \right)}}{27} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 \sin{\left({\color{red}{\left(3 x\right)}} \right)}}{27}$$

因此，

$$\int{x^{2} \cos{\left(3 x \right)} d x} = \frac{x^{2} \sin{\left(3 x \right)}}{3} + \frac{2 x \cos{\left(3 x \right)}}{9} - \frac{2 \sin{\left(3 x \right)}}{27}$$

化简：

$$\int{x^{2} \cos{\left(3 x \right)} d x} = \frac{9 x^{2} \sin{\left(3 x \right)} + 6 x \cos{\left(3 x \right)} - 2 \sin{\left(3 x \right)}}{27}$$

加上积分常数：

$$\int{x^{2} \cos{\left(3 x \right)} d x} = \frac{9 x^{2} \sin{\left(3 x \right)} + 6 x \cos{\left(3 x \right)} - 2 \sin{\left(3 x \right)}}{27}+C$$

$$$\int x^{2} \cos{\left(3 x \right)}\, dx = \frac{9 x^{2} \sin{\left(3 x \right)} + 6 x \cos{\left(3 x \right)} - 2 \sin{\left(3 x \right)}}{27} + C$$$A