$$$- 4 x + \frac{3}{x^{21}}$$$ 的積分

求$$$\int \left(- 4 x + \frac{3}{x^{21}}\right)\, dx$$$。

逐項積分：

$${\color{red}{\int{\left(- 4 x + \frac{3}{x^{21}}\right)d x}}} = {\color{red}{\left(\int{\frac{3}{x^{21}} d x} - \int{4 x d x}\right)}}$$

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

$$\int{\frac{3}{x^{21}} d x} - {\color{red}{\int{4 x d x}}} = \int{\frac{3}{x^{21}} d x} - {\color{red}{\left(4 \int{x d x}\right)}}$$

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

$$\int{\frac{3}{x^{21}} d x} - 4 {\color{red}{\int{x d x}}}=\int{\frac{3}{x^{21}} d x} - 4 {\color{red}{\frac{x^{1 + 1}}{1 + 1}}}=\int{\frac{3}{x^{21}} d x} - 4 {\color{red}{\left(\frac{x^{2}}{2}\right)}}$$

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=3$$$ 與 $$$f{\left(x \right)} = \frac{1}{x^{21}}$$$：

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

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=-21$$$：

$$- 2 x^{2} + 3 {\color{red}{\int{\frac{1}{x^{21}} d x}}}=- 2 x^{2} + 3 {\color{red}{\int{x^{-21} d x}}}=- 2 x^{2} + 3 {\color{red}{\frac{x^{-21 + 1}}{-21 + 1}}}=- 2 x^{2} + 3 {\color{red}{\left(- \frac{x^{-20}}{20}\right)}}=- 2 x^{2} + 3 {\color{red}{\left(- \frac{1}{20 x^{20}}\right)}}$$

因此，

$$\int{\left(- 4 x + \frac{3}{x^{21}}\right)d x} = - 2 x^{2} - \frac{3}{20 x^{20}}$$

化簡：

$$\int{\left(- 4 x + \frac{3}{x^{21}}\right)d x} = \frac{- 40 x^{22} - 3}{20 x^{20}}$$

加上積分常數：

$$\int{\left(- 4 x + \frac{3}{x^{21}}\right)d x} = \frac{- 40 x^{22} - 3}{20 x^{20}}+C$$

$$$\int \left(- 4 x + \frac{3}{x^{21}}\right)\, dx = \frac{- 40 x^{22} - 3}{20 x^{20}} + C$$$A