$$$\frac{1}{6 x^{7}}$$$ 的積分

求$$$\int \frac{1}{6 x^{7}}\, dx$$$。

套用常數倍法則 $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$，使用 $$$c=\frac{1}{6}$$$ 與 $$$f{\left(x \right)} = \frac{1}{x^{7}}$$$：

$${\color{red}{\int{\frac{1}{6 x^{7}} d x}}} = {\color{red}{\left(\frac{\int{\frac{1}{x^{7}} d x}}{6}\right)}}$$

套用冪次法則 $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$，以 $$$n=-7$$$：

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

因此，

$$\int{\frac{1}{6 x^{7}} d x} = - \frac{1}{36 x^{6}}$$

加上積分常數：

$$\int{\frac{1}{6 x^{7}} d x} = - \frac{1}{36 x^{6}}+C$$

$$$\int \frac{1}{6 x^{7}}\, dx = - \frac{1}{36 x^{6}} + C$$$A