Integral of $$$\frac{- 21 x - 20}{4 x^{8}}$$$

Find $$$\int \frac{- 21 x - 20}{4 x^{8}}\, dx$$$.

The input is rewritten: $$$\int{\frac{- 21 x - 20}{4 x^{8}} d x}=\int{\frac{- \frac{21 x}{4} - 5}{x^{8}} d x}$$$.

Simplify the integrand:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=-1$$$ and $$$f{\left(x \right)} = \frac{\frac{21 x}{4} + 5}{x^{8}}$$$:

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

Simplify:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=\frac{1}{4}$$$ and $$$f{\left(x \right)} = \frac{21 x + 20}{x^{8}}$$$:

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

Expand the expression:

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

Integrate term by term:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=20$$$ and $$$f{\left(x \right)} = \frac{1}{x^{8}}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-8$$$:

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

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=21$$$ and $$$f{\left(x \right)} = \frac{1}{x^{7}}$$$:

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

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-7$$$:

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

Therefore,

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

Simplify:

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

Add the constant of integration:

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

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