$$$x \cos{\left(9 x \right)}$$$ 的积分

求$$$\int x \cos{\left(9 x \right)}\, dx$$$。

对于积分$$$\int{x \cos{\left(9 x \right)} d x}$$$，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 $$$\operatorname{dv}=\cos{\left(9 x \right)} dx$$$。

则 $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{\cos{\left(9 x \right)} d x}=\frac{\sin{\left(9 x \right)}}{9}$$$ (步骤见 »)。

积分变为

$${\color{red}{\int{x \cos{\left(9 x \right)} d x}}}={\color{red}{\left(x \cdot \frac{\sin{\left(9 x \right)}}{9}-\int{\frac{\sin{\left(9 x \right)}}{9} \cdot 1 d x}\right)}}={\color{red}{\left(\frac{x \sin{\left(9 x \right)}}{9} - \int{\frac{\sin{\left(9 x \right)}}{9} d x}\right)}}$$

对 $$$c=\frac{1}{9}$$$ 和 $$$f{\left(x \right)} = \sin{\left(9 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{x \sin{\left(9 x \right)}}{9} - {\color{red}{\int{\frac{\sin{\left(9 x \right)}}{9} d x}}} = \frac{x \sin{\left(9 x \right)}}{9} - {\color{red}{\left(\frac{\int{\sin{\left(9 x \right)} d x}}{9}\right)}}$$

设$$$u=9 x$$$。

则$$$du=\left(9 x\right)^{\prime }dx = 9 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{9}$$$。

该积分可以改写为

$$\frac{x \sin{\left(9 x \right)}}{9} - \frac{{\color{red}{\int{\sin{\left(9 x \right)} d x}}}}{9} = \frac{x \sin{\left(9 x \right)}}{9} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{9} d u}}}}{9}$$

对 $$$c=\frac{1}{9}$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{x \sin{\left(9 x \right)}}{9} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{9} d u}}}}{9} = \frac{x \sin{\left(9 x \right)}}{9} - \frac{{\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{9}\right)}}}{9}$$

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

$$\frac{x \sin{\left(9 x \right)}}{9} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{81} = \frac{x \sin{\left(9 x \right)}}{9} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{81}$$

回忆一下 $$$u=9 x$$$:

$$\frac{x \sin{\left(9 x \right)}}{9} + \frac{\cos{\left({\color{red}{u}} \right)}}{81} = \frac{x \sin{\left(9 x \right)}}{9} + \frac{\cos{\left({\color{red}{\left(9 x\right)}} \right)}}{81}$$

因此，

$$\int{x \cos{\left(9 x \right)} d x} = \frac{x \sin{\left(9 x \right)}}{9} + \frac{\cos{\left(9 x \right)}}{81}$$

加上积分常数：

$$\int{x \cos{\left(9 x \right)} d x} = \frac{x \sin{\left(9 x \right)}}{9} + \frac{\cos{\left(9 x \right)}}{81}+C$$

$$$\int x \cos{\left(9 x \right)}\, dx = \left(\frac{x \sin{\left(9 x \right)}}{9} + \frac{\cos{\left(9 x \right)}}{81}\right) + C$$$A