$$$x \sin{\left(5 \right)} \cos{\left(x \right)}$$$ 的积分

求$$$\int x \sin{\left(5 \right)} \cos{\left(x \right)}\, dx$$$。

对 $$$c=\sin{\left(5 \right)}$$$ 和 $$$f{\left(x \right)} = x \cos{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{x \sin{\left(5 \right)} \cos{\left(x \right)} d x}}} = {\color{red}{\sin{\left(5 \right)} \int{x \cos{\left(x \right)} d x}}}$$

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

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

所以，

$$\sin{\left(5 \right)} {\color{red}{\int{x \cos{\left(x \right)} d x}}}=\sin{\left(5 \right)} {\color{red}{\left(x \cdot \sin{\left(x \right)}-\int{\sin{\left(x \right)} \cdot 1 d x}\right)}}=\sin{\left(5 \right)} {\color{red}{\left(x \sin{\left(x \right)} - \int{\sin{\left(x \right)} d x}\right)}}$$

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

$$\sin{\left(5 \right)} \left(x \sin{\left(x \right)} - {\color{red}{\int{\sin{\left(x \right)} d x}}}\right) = \sin{\left(5 \right)} \left(x \sin{\left(x \right)} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}\right)$$

因此，

$$\int{x \sin{\left(5 \right)} \cos{\left(x \right)} d x} = \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(5 \right)}$$

加上积分常数：

$$\int{x \sin{\left(5 \right)} \cos{\left(x \right)} d x} = \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(5 \right)}+C$$

$$$\int x \sin{\left(5 \right)} \cos{\left(x \right)}\, dx = \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) \sin{\left(5 \right)} + C$$$A