$$$\frac{- 21 x - 20}{4 x^{8}}$$$ 的积分

求$$$\int \frac{- 21 x - 20}{4 x^{8}}\, dx$$$。

输入已重写为：$$$\int{\frac{- 21 x - 20}{4 x^{8}} d x}=\int{\frac{- \frac{21 x}{4} - 5}{x^{8}} d x}$$$。

化简被积函数:

$${\color{red}{\int{\frac{- \frac{21 x}{4} - 5}{x^{8}} d x}}} = {\color{red}{\int{\left(- \frac{\frac{21 x}{4} + 5}{x^{8}}\right)d x}}}$$

对 $$$c=-1$$$ 和 $$$f{\left(x \right)} = \frac{\frac{21 x}{4} + 5}{x^{8}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$${\color{red}{\int{\left(- \frac{\frac{21 x}{4} + 5}{x^{8}}\right)d x}}} = {\color{red}{\left(- \int{\frac{\frac{21 x}{4} + 5}{x^{8}} d x}\right)}}$$

Simplify:

$$- {\color{red}{\int{\frac{\frac{21 x}{4} + 5}{x^{8}} d x}}} = - {\color{red}{\int{\frac{21 x + 20}{4 x^{8}} d x}}}$$

对 $$$c=\frac{1}{4}$$$ 和 $$$f{\left(x \right)} = \frac{21 x + 20}{x^{8}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$- {\color{red}{\int{\frac{21 x + 20}{4 x^{8}} d x}}} = - {\color{red}{\left(\frac{\int{\frac{21 x + 20}{x^{8}} d x}}{4}\right)}}$$

Expand the expression:

$$- \frac{{\color{red}{\int{\frac{21 x + 20}{x^{8}} d x}}}}{4} = - \frac{{\color{red}{\int{\left(\frac{21}{x^{7}} + \frac{20}{x^{8}}\right)d x}}}}{4}$$

逐项积分：

$$- \frac{{\color{red}{\int{\left(\frac{21}{x^{7}} + \frac{20}{x^{8}}\right)d x}}}}{4} = - \frac{{\color{red}{\left(\int{\frac{20}{x^{8}} d x} + \int{\frac{21}{x^{7}} d x}\right)}}}{4}$$

对 $$$c=20$$$ 和 $$$f{\left(x \right)} = \frac{1}{x^{8}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$- \frac{\int{\frac{21}{x^{7}} d x}}{4} - \frac{{\color{red}{\int{\frac{20}{x^{8}} d x}}}}{4} = - \frac{\int{\frac{21}{x^{7}} d x}}{4} - \frac{{\color{red}{\left(20 \int{\frac{1}{x^{8}} d x}\right)}}}{4}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-8$$$：

$$- \frac{\int{\frac{21}{x^{7}} d x}}{4} - 5 {\color{red}{\int{\frac{1}{x^{8}} d x}}}=- \frac{\int{\frac{21}{x^{7}} d x}}{4} - 5 {\color{red}{\int{x^{-8} d x}}}=- \frac{\int{\frac{21}{x^{7}} d x}}{4} - 5 {\color{red}{\frac{x^{-8 + 1}}{-8 + 1}}}=- \frac{\int{\frac{21}{x^{7}} d x}}{4} - 5 {\color{red}{\left(- \frac{x^{-7}}{7}\right)}}=- \frac{\int{\frac{21}{x^{7}} d x}}{4} - 5 {\color{red}{\left(- \frac{1}{7 x^{7}}\right)}}$$

对 $$$c=21$$$ 和 $$$f{\left(x \right)} = \frac{1}{x^{7}}$$$ 应用常数倍法则 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$：

$$- \frac{{\color{red}{\int{\frac{21}{x^{7}} d x}}}}{4} + \frac{5}{7 x^{7}} = - \frac{{\color{red}{\left(21 \int{\frac{1}{x^{7}} d x}\right)}}}{4} + \frac{5}{7 x^{7}}$$

应用幂法则 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，其中 $$$n=-7$$$：

$$- \frac{21 {\color{red}{\int{\frac{1}{x^{7}} d x}}}}{4} + \frac{5}{7 x^{7}}=- \frac{21 {\color{red}{\int{x^{-7} d x}}}}{4} + \frac{5}{7 x^{7}}=- \frac{21 {\color{red}{\frac{x^{-7 + 1}}{-7 + 1}}}}{4} + \frac{5}{7 x^{7}}=- \frac{21 {\color{red}{\left(- \frac{x^{-6}}{6}\right)}}}{4} + \frac{5}{7 x^{7}}=- \frac{21 {\color{red}{\left(- \frac{1}{6 x^{6}}\right)}}}{4} + \frac{5}{7 x^{7}}$$

因此，

$$\int{\frac{- \frac{21 x}{4} - 5}{x^{8}} d x} = \frac{7}{8 x^{6}} + \frac{5}{7 x^{7}}$$

化简：

$$\int{\frac{- \frac{21 x}{4} - 5}{x^{8}} d x} = \frac{49 x + 40}{56 x^{7}}$$

加上积分常数：

$$\int{\frac{- \frac{21 x}{4} - 5}{x^{8}} d x} = \frac{49 x + 40}{56 x^{7}}+C$$

$$$\int \frac{- 21 x - 20}{4 x^{8}}\, dx = \frac{49 x + 40}{56 x^{7}} + C$$$A