The power rule that applies to [tex]f(x)= \frac{1}{x} [/tex] is [tex]f(x)= x^{-1} [/tex]
Integrating [tex] \int\ {x^{-1} } \, dx [/tex] will give the effect of
[tex] \frac{x^{-1+1} }{-1+1} = \frac{ x^{0} }{0} [/tex], which is undefined since we cannot divide by '0'
The conclusion is that to integrate [tex]f(x)= \frac{1}{x} [/tex] we don't use the power rule. We use instead
[tex] \int\ { \frac{1}{x} } \, dx =ln(x)[/tex]
