$$$a l t \left(x - \pi\right) \cos{\left(x \right)}$$$ 對 $$$x$$$ 的積分
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您的輸入
求$$$\int a l t \left(x - \pi\right) \cos{\left(x \right)}\, dx$$$。
解答
套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=a l t$$$ 與 $$$f{\left(x \right)} = \left(x - \pi\right) \cos{\left(x \right)}$$$：
$${\color{red}{\int{a l t \left(x - \pi\right) \cos{\left(x \right)} d x}}} = {\color{red}{a l t \int{\left(x - \pi\right) \cos{\left(x \right)} d x}}}$$
對於積分 $$$\int{\left(x - \pi\right) \cos{\left(x \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。
令 $$$\operatorname{u}=x - \pi$$$ 與 $$$\operatorname{dv}=\cos{\left(x \right)} dx$$$。
則 $$$\operatorname{du}=\left(x - \pi\right)^{\prime }dx=1 dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{\cos{\left(x \right)} d x}=\sin{\left(x \right)}$$$（步驟見 »）。
因此，
$$a l t {\color{red}{\int{\left(x - \pi\right) \cos{\left(x \right)} d x}}}=a l t {\color{red}{\left(\left(x - \pi\right) \cdot \sin{\left(x \right)}-\int{\sin{\left(x \right)} \cdot 1 d x}\right)}}=a l t {\color{red}{\left(\left(x - \pi\right) \sin{\left(x \right)} - \int{\sin{\left(x \right)} d x}\right)}}$$
正弦函數的積分為 $$$\int{\sin{\left(x \right)} d x} = - \cos{\left(x \right)}$$$：
$$a l t \left(\left(x - \pi\right) \sin{\left(x \right)} - {\color{red}{\int{\sin{\left(x \right)} d x}}}\right) = a l t \left(\left(x - \pi\right) \sin{\left(x \right)} - {\color{red}{\left(- \cos{\left(x \right)}\right)}}\right)$$
因此，
$$\int{a l t \left(x - \pi\right) \cos{\left(x \right)} d x} = a l t \left(\left(x - \pi\right) \sin{\left(x \right)} + \cos{\left(x \right)}\right)$$
化簡：
$$\int{a l t \left(x - \pi\right) \cos{\left(x \right)} d x} = - a l t \left(\left(\pi - x\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right)$$
加上積分常數：
$$\int{a l t \left(x - \pi\right) \cos{\left(x \right)} d x} = - a l t \left(\left(\pi - x\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right)+C$$
答案
$$$\int a l t \left(x - \pi\right) \cos{\left(x \right)}\, dx = - a l t \left(\left(\pi - x\right) \sin{\left(x \right)} - \cos{\left(x \right)}\right) + C$$$A