Integral of $$$\frac{5}{x^{2}}$$$

Find $$$\int \frac{5}{x^{2}}\, dx$$$.

Apply the constant multiple rule $$$\int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx$$$ with $$$c=5$$$ and $$$f{\left(x \right)} = \frac{1}{x^{2}}$$$:

$${\color{red}{\int{\frac{5}{x^{2}} d x}}} = {\color{red}{\left(5 \int{\frac{1}{x^{2}} d x}\right)}}$$

Apply the power rule $$$\int x^{n}\, dx = \frac{x^{n + 1}}{n + 1}$$$ $$$\left(n \neq -1 \right)$$$ with $$$n=-2$$$:

$$5 {\color{red}{\int{\frac{1}{x^{2}} d x}}}=5 {\color{red}{\int{x^{-2} d x}}}=5 {\color{red}{\frac{x^{-2 + 1}}{-2 + 1}}}=5 {\color{red}{\left(- x^{-1}\right)}}=5 {\color{red}{\left(- \frac{1}{x}\right)}}$$

Therefore,

$$\int{\frac{5}{x^{2}} d x} = - \frac{5}{x}$$

Add the constant of integration:

$$\int{\frac{5}{x^{2}} d x} = - \frac{5}{x}+C$$

$$$\int \frac{5}{x^{2}}\, dx = - \frac{5}{x} + C$$$A