Integral de $$$\frac{1}{a^{2} x^{2}}$$$ con respecto a $$$x$$$

Halla $$$\int \frac{1}{a^{2} x^{2}}\, dx$$$.

Aplica la regla del factor constante $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ con $$$c=\frac{1}{a^{2}}$$$ y $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:

$${\color{red}{\int{\frac{1}{a^{2} x^{2}} d x}}} = {\color{red}{\frac{\int{\frac{1}{x^{2}} d x}}{a^{2}}}}$$

Aplica la regla de la potencia $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ con $$$n=-2$$$:

$$\frac{{\color{red}{\int{\frac{1}{x^{2}} d x}}}}{a^{2}}=\frac{{\color{red}{\int{x^{-2} d x}}}}{a^{2}}=\frac{{\color{red}{\frac{x^{-2 + 1}}{-2 + 1}}}}{a^{2}}=\frac{{\color{red}{\left(- x^{-1}\right)}}}{a^{2}}=\frac{{\color{red}{\left(- \frac{1}{x}\right)}}}{a^{2}}$$

Por lo tanto,

$$\int{\frac{1}{a^{2} x^{2}} d x} = - \frac{1}{a^{2} x}$$

Añade la constante de integración:

$$\int{\frac{1}{a^{2} x^{2}} d x} = - \frac{1}{a^{2} x}+C$$

$$$\int \frac{1}{a^{2} x^{2}}\, dx = - \frac{1}{a^{2} x} + C$$$A