Integral de $$$\frac{4}{x^{8}}$$$

Halla $$$\int \frac{4}{x^{8}}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=4$$$ y $$$f{\left(x \right)} = \frac{1}{x^{8}}$$$:

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

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=-8$$$:

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

Por lo tanto,

$$\int{\frac{4}{x^{8}} d x} = - \frac{4}{7 x^{7}}$$

Añade la constante de integración:

$$\int{\frac{4}{x^{8}} d x} = - \frac{4}{7 x^{7}}+C$$

$$$\int \frac{4}{x^{8}}\, dx = - \frac{4}{7 x^{7}} + C$$$A