$$$4 x^{3} \sin{\left(x \right)}$$$ 的積分

求$$$\int 4 x^{3} \sin{\left(x \right)}\, dx$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=4$$$ 與 $$$f{\left(x \right)} = x^{3} \sin{\left(x \right)}$$$：

$${\color{red}{\int{4 x^{3} \sin{\left(x \right)} d x}}} = {\color{red}{\left(4 \int{x^{3} \sin{\left(x \right)} d x}\right)}}$$

對於積分 $$$\int{x^{3} \sin{\left(x \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

令 $$$\operatorname{u}=x^{3}$$$ 與 $$$\operatorname{dv}=\sin{\left(x \right)} dx$$$。

則 $$$\operatorname{du}=\left(x^{3}\right)^{\prime }dx=3 x^{2} dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{\sin{\left(x \right)} d x}=- \cos{\left(x \right)}$$$（步驟見 »）。

所以，

$$4 {\color{red}{\int{x^{3} \sin{\left(x \right)} d x}}}=4 {\color{red}{\left(x^{3} \cdot \left(- \cos{\left(x \right)}\right)-\int{\left(- \cos{\left(x \right)}\right) \cdot 3 x^{2} d x}\right)}}=4 {\color{red}{\left(- x^{3} \cos{\left(x \right)} - \int{\left(- 3 x^{2} \cos{\left(x \right)}\right)d x}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=-3$$$ 與 $$$f{\left(x \right)} = x^{2} \cos{\left(x \right)}$$$：

$$- 4 x^{3} \cos{\left(x \right)} - 4 {\color{red}{\int{\left(- 3 x^{2} \cos{\left(x \right)}\right)d x}}} = - 4 x^{3} \cos{\left(x \right)} - 4 {\color{red}{\left(- 3 \int{x^{2} \cos{\left(x \right)} d x}\right)}}$$

對於積分 $$$\int{x^{2} \cos{\left(x \right)} d x}$$$，使用分部積分法 $$$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$$$。

令 $$$\operatorname{u}=x^{2}$$$ 與 $$$\operatorname{dv}=\cos{\left(x \right)} dx$$$。

則 $$$\operatorname{du}=\left(x^{2}\right)^{\prime }dx=2 x dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{\cos{\left(x \right)} d x}=\sin{\left(x \right)}$$$（步驟見 »）。

該積分變為

$$- 4 x^{3} \cos{\left(x \right)} + 12 {\color{red}{\int{x^{2} \cos{\left(x \right)} d x}}}=- 4 x^{3} \cos{\left(x \right)} + 12 {\color{red}{\left(x^{2} \cdot \sin{\left(x \right)}-\int{\sin{\left(x \right)} \cdot 2 x d x}\right)}}=- 4 x^{3} \cos{\left(x \right)} + 12 {\color{red}{\left(x^{2} \sin{\left(x \right)} - \int{2 x \sin{\left(x \right)} d x}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=2$$$ 與 $$$f{\left(x \right)} = x \sin{\left(x \right)}$$$：

$$- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} - 12 {\color{red}{\int{2 x \sin{\left(x \right)} d x}}} = - 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} - 12 {\color{red}{\left(2 \int{x \sin{\left(x \right)} d x}\right)}}$$

對於積分 $$$\int{x \sin{\left(x \right)} d x}$$$，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 $$$\operatorname{dv}=\sin{\left(x \right)} dx$$$。

則 $$$\operatorname{du}=\left(x\right)^{\prime }dx=1 dx$$$（步驟見 »），且 $$$\operatorname{v}=\int{\sin{\left(x \right)} d x}=- \cos{\left(x \right)}$$$（步驟見 »）。

所以，

$$- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} - 24 {\color{red}{\int{x \sin{\left(x \right)} d x}}}=- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} - 24 {\color{red}{\left(x \cdot \left(- \cos{\left(x \right)}\right)-\int{\left(- \cos{\left(x \right)}\right) \cdot 1 d x}\right)}}=- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} - 24 {\color{red}{\left(- x \cos{\left(x \right)} - \int{\left(- \cos{\left(x \right)}\right)d x}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=-1$$$ 與 $$$f{\left(x \right)} = \cos{\left(x \right)}$$$：

$$- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} + 24 {\color{red}{\int{\left(- \cos{\left(x \right)}\right)d x}}} = - 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} + 24 {\color{red}{\left(- \int{\cos{\left(x \right)} d x}\right)}}$$

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

$$- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 {\color{red}{\int{\cos{\left(x \right)} d x}}} = - 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 {\color{red}{\sin{\left(x \right)}}}$$

因此，

$$\int{4 x^{3} \sin{\left(x \right)} d x} = - 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 \sin{\left(x \right)}$$

加上積分常數：

$$\int{4 x^{3} \sin{\left(x \right)} d x} = - 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 \sin{\left(x \right)}+C$$

$$$\int 4 x^{3} \sin{\left(x \right)}\, dx = \left(- 4 x^{3} \cos{\left(x \right)} + 12 x^{2} \sin{\left(x \right)} + 24 x \cos{\left(x \right)} - 24 \sin{\left(x \right)}\right) + C$$$A