$$$\frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}}$$$ 关于$$$x$$$的积分

求$$$\int \frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}}\, dx$$$。

对 $$$c=\frac{s^{2} \sin{\left(1 \right)}}{c_{0}}$$$ 和 $$$f{\left(x \right)} = x \cos{\left(2 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{\frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}} d x}}} = {\color{red}{\frac{s^{2} \sin{\left(1 \right)} \int{x \cos{\left(2 x \right)} d x}}{c_{0}}}}$$

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

则 $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{\cos{\left(2 x \right)} d x}=\frac{\sin{\left(2 x \right)}}{2}$$$ (步骤见 »)。

所以，

$$\frac{s^{2} \sin{\left(1 \right)} {\color{red}{\int{x \cos{\left(2 x \right)} d x}}}}{c_{0}}=\frac{s^{2} \sin{\left(1 \right)} {\color{red}{\left(x \cdot \frac{\sin{\left(2 x \right)}}{2}-\int{\frac{\sin{\left(2 x \right)}}{2} \cdot 1 d x}\right)}}}{c_{0}}=\frac{s^{2} \sin{\left(1 \right)} {\color{red}{\left(\frac{x \sin{\left(2 x \right)}}{2} - \int{\frac{\sin{\left(2 x \right)}}{2} d x}\right)}}}{c_{0}}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = \sin{\left(2 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$\frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - {\color{red}{\int{\frac{\sin{\left(2 x \right)}}{2} d x}}}\right)}{c_{0}} = \frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - {\color{red}{\left(\frac{\int{\sin{\left(2 x \right)} d x}}{2}\right)}}\right)}{c_{0}}$$

设$$$u=2 x$$$。

则$$$du=\left(2 x\right)^{\prime }dx = 2 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{2}$$$。

因此，

$$\frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\int{\sin{\left(2 x \right)} d x}}}}{2}\right)}{c_{0}} = \frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}}{2}\right)}{c_{0}}$$

对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$\frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\int{\frac{\sin{\left(u \right)}}{2} d u}}}}{2}\right)}{c_{0}} = \frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\left(\frac{\int{\sin{\left(u \right)} d u}}{2}\right)}}}{2}\right)}{c_{0}}$$

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

$$\frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{4}\right)}{c_{0}} = \frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{4}\right)}{c_{0}}$$

回忆一下 $$$u=2 x$$$:

$$\frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left({\color{red}{u}} \right)}}{4}\right)}{c_{0}} = \frac{s^{2} \sin{\left(1 \right)} \left(\frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left({\color{red}{\left(2 x\right)}} \right)}}{4}\right)}{c_{0}}$$

因此，

$$\int{\frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}} d x} = \frac{s^{2} \left(\frac{x \sin{\left(2 x \right)}}{2} + \frac{\cos{\left(2 x \right)}}{4}\right) \sin{\left(1 \right)}}{c_{0}}$$

化简：

$$\int{\frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}} d x} = \frac{s^{2} \left(2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) \sin{\left(1 \right)}}{4 c_{0}}$$

加上积分常数：

$$\int{\frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}} d x} = \frac{s^{2} \left(2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) \sin{\left(1 \right)}}{4 c_{0}}+C$$

$$$\int \frac{s^{2} x \sin{\left(1 \right)} \cos{\left(2 x \right)}}{c_{0}}\, dx = \frac{s^{2} \left(2 x \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) \sin{\left(1 \right)}}{4 c_{0}} + C$$$A