$$$x e^{6} \sin{\left(7 x \right)}$$$ 的积分

求$$$\int x e^{6} \sin{\left(7 x \right)}\, dx$$$。

对 $$$c=e^{6}$$$ 和 $$$f{\left(x \right)} = x \sin{\left(7 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{x e^{6} \sin{\left(7 x \right)} d x}}} = {\color{red}{e^{6} \int{x \sin{\left(7 x \right)} d x}}}$$

对于积分$$$\int{x \sin{\left(7 x \right)} d x}$$$，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 $$$\operatorname{dv}=\sin{\left(7 x \right)} dx$$$。

则 $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$ (步骤见 »)，并且 $$$\operatorname{v}=\int{\sin{\left(7 x \right)} d x}=- \frac{\cos{\left(7 x \right)}}{7}$$$ (步骤见 »)。

所以，

$$e^{6} {\color{red}{\int{x \sin{\left(7 x \right)} d x}}}=e^{6} {\color{red}{\left(x \cdot \left(- \frac{\cos{\left(7 x \right)}}{7}\right)-\int{\left(- \frac{\cos{\left(7 x \right)}}{7}\right) \cdot 1 d x}\right)}}=e^{6} {\color{red}{\left(- \frac{x \cos{\left(7 x \right)}}{7} - \int{\left(- \frac{\cos{\left(7 x \right)}}{7}\right)d x}\right)}}$$

对 $$$c=- \frac{1}{7}$$$ 和 $$$f{\left(x \right)} = \cos{\left(7 x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$e^{6} \left(- \frac{x \cos{\left(7 x \right)}}{7} - {\color{red}{\int{\left(- \frac{\cos{\left(7 x \right)}}{7}\right)d x}}}\right) = e^{6} \left(- \frac{x \cos{\left(7 x \right)}}{7} - {\color{red}{\left(- \frac{\int{\cos{\left(7 x \right)} d x}}{7}\right)}}\right)$$

设$$$u=7 x$$$。

则$$$du=\left(7 x\right)^{\prime }dx = 7 dx$$$ (步骤见»)，并有$$$dx = \frac{du}{7}$$$。

积分变为

$$e^{6} \left(- \frac{x \cos{\left(7 x \right)}}{7} + \frac{{\color{red}{\int{\cos{\left(7 x \right)} d x}}}}{7}\right) = e^{6} \left(- \frac{x \cos{\left(7 x \right)}}{7} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}}}{7}\right)$$

对 $$$c=\frac{1}{7}$$$ 和 $$$f{\left(u \right)} = \cos{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

$$e^{6} \left(- \frac{x \cos{\left(7 x \right)}}{7} + \frac{{\color{red}{\int{\frac{\cos{\left(u \right)}}{7} d u}}}}{7}\right) = e^{6} \left(- \frac{x \cos{\left(7 x \right)}}{7} + \frac{{\color{red}{\left(\frac{\int{\cos{\left(u \right)} d u}}{7}\right)}}}{7}\right)$$

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

$$e^{6} \left(- \frac{x \cos{\left(7 x \right)}}{7} + \frac{{\color{red}{\int{\cos{\left(u \right)} d u}}}}{49}\right) = e^{6} \left(- \frac{x \cos{\left(7 x \right)}}{7} + \frac{{\color{red}{\sin{\left(u \right)}}}}{49}\right)$$

回忆一下 $$$u=7 x$$$:

$$e^{6} \left(- \frac{x \cos{\left(7 x \right)}}{7} + \frac{\sin{\left({\color{red}{u}} \right)}}{49}\right) = e^{6} \left(- \frac{x \cos{\left(7 x \right)}}{7} + \frac{\sin{\left({\color{red}{\left(7 x\right)}} \right)}}{49}\right)$$

因此，

$$\int{x e^{6} \sin{\left(7 x \right)} d x} = \left(- \frac{x \cos{\left(7 x \right)}}{7} + \frac{\sin{\left(7 x \right)}}{49}\right) e^{6}$$

化简：

$$\int{x e^{6} \sin{\left(7 x \right)} d x} = \frac{\left(- 7 x \cos{\left(7 x \right)} + \sin{\left(7 x \right)}\right) e^{6}}{49}$$

加上积分常数：

$$\int{x e^{6} \sin{\left(7 x \right)} d x} = \frac{\left(- 7 x \cos{\left(7 x \right)} + \sin{\left(7 x \right)}\right) e^{6}}{49}+C$$

$$$\int x e^{6} \sin{\left(7 x \right)}\, dx = \frac{\left(- 7 x \cos{\left(7 x \right)} + \sin{\left(7 x \right)}\right) e^{6}}{49} + C$$$A