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$$$x \cos{\left(x \right)}$$$ 的積分
您的輸入
求$$$\int x \cos{\left(x \right)}\, dx$$$。
解答
對於積分 $$$\int{x \cos{\left(x \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。
令 $$$\operatorname{u}=x$$$ 與 $$$\operatorname{dv}=\cos{\left(x \right)} dx$$$。
則 $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{\cos{\left(x \right)} d x}=\sin{\left(x \right)}$$$（步驟見 »）。
該積分可改寫為
$${\color{red}{\int{x \cos{\left(x \right)} d x}}}={\color{red}{\left(x \cdot \sin{\left(x \right)}-\int{\sin{\left(x \right)} \cdot 1 d x}\right)}}={\color{red}{\left(x \sin{\left(x \right)} - \int{\sin{\left(x \right)} d x}\right)}}$$
正弦函數的積分為 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$：
$$x \sin{\left(x \right)} - {\color{red}{\int{\sin{\left(x \right)} d x}}} = x \sin{\left(x \right)} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}$$
因此，
$$\int{x \cos{\left(x \right)} d x} = x \sin{\left(x \right)} + \cos{\left(x \right)}$$
加上積分常數：
$$\int{x \cos{\left(x \right)} d x} = x \sin{\left(x \right)} + \cos{\left(x \right)}+C$$
答案
$$$\int x \cos{\left(x \right)}\, dx = \left(x \sin{\left(x \right)} + \cos{\left(x \right)}\right) + C$$$A