$$$\frac{- 21 x - 20}{4 x^{8}}$$$の積分

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

入力は次のように書き換えられます: $$$\int{\frac{- 21 x - 20}{4 x^{8}} d x}=\int{\frac{- \frac{21 x}{4} - 5}{x^{8}} d x}$$$。

被積分関数を簡単化する:

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

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=-1$$$ と $$$f{\left(x \right)} = \frac{\frac{21 x}{4} + 5}{x^{8}}$$$ に対して適用する:

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

Simplify:

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

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=\frac{1}{4}$$$ と $$$f{\left(x \right)} = \frac{21 x + 20}{x^{8}}$$$ に対して適用する:

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

Expand the expression:

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

項別に積分せよ:

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

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=20$$$ と $$$f{\left(x \right)} = \frac{1}{x^{8}}$$$ に対して適用する:

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

$$$n=-8$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:

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

定数倍の法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ を、$$$c=21$$$ と $$$f{\left(x \right)} = \frac{1}{x^{7}}$$$ に対して適用する:

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

$$$n=-7$$$ を用いて、べき乗の法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ を適用します:

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

したがって、

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

簡単化せよ:

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

積分定数を加える:

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

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