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

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

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

$${\color{red}{\int{x e^{3} \cos{\left(x e^{3} \right)} d x}}} = {\color{red}{e^{3} \int{x \cos{\left(x e^{3} \right)} d x}}}$$

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

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

所以，

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

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

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

设$$$u=x e^{3}$$$。

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

所以，

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

对 $$$c=e^{-3}$$$ 和 $$$f{\left(u \right)} = \sin{\left(u \right)}$$$ 应用常数倍法则 $$$\int c f{\left(u \right)}\, du = c \int f{\left(u \right)}\, du$$$：

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

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

$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\int{\sin{\left(u \right)} d u}}}}{e^{6}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} - \frac{{\color{red}{\left(- \cos{\left(u \right)}\right)}}}{e^{6}}\right)$$

回忆一下 $$$u=x e^{3}$$$:

$$e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} + \frac{\cos{\left({\color{red}{u}} \right)}}{e^{6}}\right) = e^{3} \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} + \frac{\cos{\left({\color{red}{x e^{3}}} \right)}}{e^{6}}\right)$$

因此，

$$\int{x e^{3} \cos{\left(x e^{3} \right)} d x} = \left(\frac{x \sin{\left(x e^{3} \right)}}{e^{3}} + \frac{\cos{\left(x e^{3} \right)}}{e^{6}}\right) e^{3}$$

化简：

$$\int{x e^{3} \cos{\left(x e^{3} \right)} d x} = x \sin{\left(x e^{3} \right)} + \frac{\cos{\left(x e^{3} \right)}}{e^{3}}$$

加上积分常数：

$$\int{x e^{3} \cos{\left(x e^{3} \right)} d x} = x \sin{\left(x e^{3} \right)} + \frac{\cos{\left(x e^{3} \right)}}{e^{3}}+C$$

$$$\int x e^{3} \cos{\left(x e^{3} \right)}\, dx = \left(x \sin{\left(x e^{3} \right)} + \frac{\cos{\left(x e^{3} \right)}}{e^{3}}\right) + C$$$A