$$$x \left(21 x - 21\right) e^{2} - \left(31 x - 31\right) \left(x e^{2} - 4\right)$$$ 的積分

求$$$\int \left(x \left(21 x - 21\right) e^{2} - \left(31 x - 31\right) \left(x e^{2} - 4\right)\right)\, dx$$$。

逐項積分：

$${\color{red}{\int{\left(x \left(21 x - 21\right) e^{2} - \left(31 x - 31\right) \left(x e^{2} - 4\right)\right)d x}}} = {\color{red}{\left(- \int{\left(31 x - 31\right) \left(x e^{2} - 4\right) d x} + \int{x \left(21 x - 21\right) e^{2} d x}\right)}}$$

簡化被積函數:

$$\int{x \left(21 x - 21\right) e^{2} d x} - {\color{red}{\int{\left(31 x - 31\right) \left(x e^{2} - 4\right) d x}}} = \int{x \left(21 x - 21\right) e^{2} d x} - {\color{red}{\int{31 \left(x - 1\right) \left(x e^{2} - 4\right) d x}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=31$$$ 與 $$$f{\left(x \right)} = \left(x - 1\right) \left(x e^{2} - 4\right)$$$：

$$\int{x \left(21 x - 21\right) e^{2} d x} - {\color{red}{\int{31 \left(x - 1\right) \left(x e^{2} - 4\right) d x}}} = \int{x \left(21 x - 21\right) e^{2} d x} - {\color{red}{\left(31 \int{\left(x - 1\right) \left(x e^{2} - 4\right) d x}\right)}}$$

Expand the expression:

$$\int{x \left(21 x - 21\right) e^{2} d x} - 31 {\color{red}{\int{\left(x - 1\right) \left(x e^{2} - 4\right) d x}}} = \int{x \left(21 x - 21\right) e^{2} d x} - 31 {\color{red}{\int{\left(x^{2} e^{2} - x e^{2} - 4 x + 4\right)d x}}}$$

逐項積分：

$$\int{x \left(21 x - 21\right) e^{2} d x} - 31 {\color{red}{\int{\left(x^{2} e^{2} - x e^{2} - 4 x + 4\right)d x}}} = \int{x \left(21 x - 21\right) e^{2} d x} - 31 {\color{red}{\left(\int{4 d x} - \int{4 x d x} - \int{x e^{2} d x} + \int{x^{2} e^{2} d x}\right)}}$$

配合 $$$c=4$$$，應用常數法則 $$$\int c\, dx = c x$$$：

$$31 \int{4 x d x} + 31 \int{x e^{2} d x} - 31 \int{x^{2} e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} - 31 {\color{red}{\int{4 d x}}} = 31 \int{4 x d x} + 31 \int{x e^{2} d x} - 31 \int{x^{2} e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} - 31 {\color{red}{\left(4 x\right)}}$$

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

$$- 124 x + 31 \int{x e^{2} d x} - 31 \int{x^{2} e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} + 31 {\color{red}{\int{4 x d x}}} = - 124 x + 31 \int{x e^{2} d x} - 31 \int{x^{2} e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} + 31 {\color{red}{\left(4 \int{x d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=1$$$：

$$- 124 x + 31 \int{x e^{2} d x} - 31 \int{x^{2} e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} + 124 {\color{red}{\int{x d x}}}=- 124 x + 31 \int{x e^{2} d x} - 31 \int{x^{2} e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} + 124 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- 124 x + 31 \int{x e^{2} d x} - 31 \int{x^{2} e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} + 124 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

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

$$62 x^{2} - 124 x + 31 \int{x e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} - 31 {\color{red}{\int{x^{2} e^{2} d x}}} = 62 x^{2} - 124 x + 31 \int{x e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} - 31 {\color{red}{e^{2} \int{x^{2} d x}}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=2$$$：

$$62 x^{2} - 124 x + 31 \int{x e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} - 31 e^{2} {\color{red}{\int{x^{2} d x}}}=62 x^{2} - 124 x + 31 \int{x e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} - 31 e^{2} {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}=62 x^{2} - 124 x + 31 \int{x e^{2} d x} + \int{x \left(21 x - 21\right) e^{2} d x} - 31 e^{2} {\color{red}{\left(\frac{x^{3}}{3}\right)}}$$

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

$$- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} - 124 x + \int{x \left(21 x - 21\right) e^{2} d x} + 31 {\color{red}{\int{x e^{2} d x}}} = - \frac{31 x^{3} e^{2}}{3} + 62 x^{2} - 124 x + \int{x \left(21 x - 21\right) e^{2} d x} + 31 {\color{red}{e^{2} \int{x d x}}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=1$$$：

$$- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} - 124 x + \int{x \left(21 x - 21\right) e^{2} d x} + 31 e^{2} {\color{red}{\int{x d x}}}=- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} - 124 x + \int{x \left(21 x - 21\right) e^{2} d x} + 31 e^{2} {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} - 124 x + \int{x \left(21 x - 21\right) e^{2} d x} + 31 e^{2} {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

簡化被積函數:

$$- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + {\color{red}{\int{x \left(21 x - 21\right) e^{2} d x}}} = - \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + {\color{red}{\int{21 x \left(x - 1\right) e^{2} d x}}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=21 e^{2}$$$ 與 $$$f{\left(x \right)} = x \left(x - 1\right)$$$：

$$- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + {\color{red}{\int{21 x \left(x - 1\right) e^{2} d x}}} = - \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + {\color{red}{\left(21 e^{2} \int{x \left(x - 1\right) d x}\right)}}$$

Expand the expression:

$$- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 e^{2} {\color{red}{\int{x \left(x - 1\right) d x}}} = - \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 e^{2} {\color{red}{\int{\left(x^{2} - x\right)d x}}}$$

逐項積分：

$$- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 e^{2} {\color{red}{\int{\left(x^{2} - x\right)d x}}} = - \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 e^{2} {\color{red}{\left(- \int{x d x} + \int{x^{2} d x}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=2$$$：

$$- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 e^{2} \left(- \int{x d x} + {\color{red}{\int{x^{2} d x}}}\right)=- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 e^{2} \left(- \int{x d x} + {\color{red}{\frac{x^{1 + 2}}{1 + 2}}}\right)=- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 e^{2} \left(- \int{x d x} + {\color{red}{\left(\frac{x^{3}}{3}\right)}}\right)$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=1$$$：

$$- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 e^{2} \left(\frac{x^{3}}{3} - {\color{red}{\int{x d x}}}\right)=- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 e^{2} \left(\frac{x^{3}}{3} - {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}\right)=- \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 e^{2} \left(\frac{x^{3}}{3} - {\color{red}{\left(\frac{x^{2}}{2}\right)}}\right)$$

因此，

$$\int{\left(x \left(21 x - 21\right) e^{2} - \left(31 x - 31\right) \left(x e^{2} - 4\right)\right)d x} = - \frac{31 x^{3} e^{2}}{3} + 62 x^{2} + \frac{31 x^{2} e^{2}}{2} - 124 x + 21 \left(\frac{x^{3}}{3} - \frac{x^{2}}{2}\right) e^{2}$$

化簡：

$$\int{\left(x \left(21 x - 21\right) e^{2} - \left(31 x - 31\right) \left(x e^{2} - 4\right)\right)d x} = \frac{x \left(- 10 x^{2} e^{2} + 15 x e^{2} + 186 x - 372\right)}{3}$$

加上積分常數：

$$\int{\left(x \left(21 x - 21\right) e^{2} - \left(31 x - 31\right) \left(x e^{2} - 4\right)\right)d x} = \frac{x \left(- 10 x^{2} e^{2} + 15 x e^{2} + 186 x - 372\right)}{3}+C$$

$$$\int \left(x \left(21 x - 21\right) e^{2} - \left(31 x - 31\right) \left(x e^{2} - 4\right)\right)\, dx = \frac{x \left(- 10 x^{2} e^{2} + 15 x e^{2} + 186 x - 372\right)}{3} + C$$$A