$$$\frac{x \cos{\left(x \right)}}{2}$$$ 的积分
您的输入
求$$$\int \frac{x \cos{\left(x \right)}}{2}\, dx$$$。
解答
对 $$$c=\frac{1}{2}$$$ 和 $$$f{\left(x \right)} = x \cos{\left(x \right)}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：
$${\color{red}{\int{\frac{x \cos{\left(x \right)}}{2} d x}}} = {\color{red}{\left(\frac{\int{x \cos{\left(x \right)} d x}}{2}\right)}}$$
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 »)。
因此，
$$\frac{{\color{red}{\int{x \cos{\left(x \right)} d x}}}}{2}=\frac{{\color{red}{\left(x \cdot \sin{\left(x \right)}-\int{\sin{\left(x \right)} \cdot 1 d x}\right)}}}{2}=\frac{{\color{red}{\left(x \sin{\left(x \right)} - \int{\sin{\left(x \right)} d x}\right)}}}{2}$$
正弦函数的积分为 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$:
$$\frac{x \sin{\left(x \right)}}{2} - \frac{{\color{red}{\int{\sin{\left(x \right)} d x}}}}{2} = \frac{x \sin{\left(x \right)}}{2} - \frac{{\color{red}{\left(- \cos{\left(x \right)}\right)}}}{2}$$
因此，
$$\int{\frac{x \cos{\left(x \right)}}{2} d x} = \frac{x \sin{\left(x \right)}}{2} + \frac{\cos{\left(x \right)}}{2}$$
化简：
$$\int{\frac{x \cos{\left(x \right)}}{2} d x} = \frac{x \sin{\left(x \right)} + \cos{\left(x \right)}}{2}$$
加上积分常数：
$$\int{\frac{x \cos{\left(x \right)}}{2} d x} = \frac{x \sin{\left(x \right)} + \cos{\left(x \right)}}{2}+C$$
答案
$$$\int \frac{x \cos{\left(x \right)}}{2}\, dx = \frac{x \sin{\left(x \right)} + \cos{\left(x \right)}}{2} + C$$$A