The deep reason why $\int \frac{1}{x}\operatorname{d}x$ is a transcendental function ($\log$) In general, the indefinite integral of $x^n$  has power $n+1$.  This is the standard power rule.  Why does it "break" for $n=-1$?  In other words, the derivative rule $$\frac{d}{dx} x^{n} = nx^{n-1}$$ fails to hold for $n=0$.
Is there some deep reason for this discontinuity?

The paradox disappears when you consider the antiderivatives having the common point $(1,0)$, i.e. the functions
$$\int_1^x t^{\alpha-1}dt=\frac{x^\alpha-1}\alpha,$$
which are plotted below for exponents $-\dfrac12,-\dfrac14,-\dfrac18,0,\dfrac18,\dfrac14,\dfrac12$.
The blue curve corresponds to $\alpha=0$ but is evaluated as the limit
$$\lim_{\alpha\to0}\frac{x^\alpha-1}\alpha=\ln(x).$$
It perfectly blends with the others.

