$$$x \sin{\left(5 \right)} \cos{\left(x \right)}$$$ 的積分

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

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\sin{\left(5 \right)}$$$ 與 $$$f{\left(x \right)} = x \cos{\left(x \right)}$$$：

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

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 »）。

所以，

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

正弦函數的積分為 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$：

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

因此，

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

加上積分常數：

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

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