Analyticity of $n$-th root function

Question

I started studying complex integrals and some doubts arose. Among them is how to solve the following one, for example:

$$\displaystyle\int_A^B \frac{x^{\frac{1-n}{n}}}{n}dx $$

Which clearly, its antiderivative is the $n$-th root. However, I have read that it is only possible to use the Fundamental Theorem of Calculus if the integrand is analytic. Otherwise, I should calculate using paths. How should I proceed? Am I correct about using paths? I'm very confused on this. Where A and B are complex.

EDIT

After reading the comments, I am attempted to think I understood this. Let me explain:

The Fundamental Theorem of Calculus will only work if the integrand is analytic/holomorphic (which I do not know exactly what it means). Otherwise, if it is not, then we will have to integrate it through a path (which I do not know how to set a path or choose).

EDIT II

So, basically, if both $A,B$ are in $-\dfrac{\pi}{2} < arg(z) < \dfrac{\pi}{2}$, the expression. becomes:

$$\sqrt[n]{B} - \sqrt[n]{A}$$?

Also, generally, if $g$ is analytic in a connected domain:

$$\displaystyle\int_A^B \frac{dg(z)}{dz} dz = g(B) - g(A)$$

Answer 1

The first problem is, as @Lubin commented, what does $x^{1/n-1}$ mean if $x$ is complex. There are $n$ $n$th roots of a complex number. Now it is customary to chose the main branch, that is, the root with the smallest angle to the real axis, with a preference to the positive angle.

It may happen that this introduces a jump along the direct path $x(t)=A+t(B-A)$, $t\in[0,1]$, from $A$ to $B$. Using a (differentiable) path $x:[0,1]\to\Bbb C$, $x(0)=A$, $x(1)=B$, that avoids the negative real half-axis results in a continuous integrand. But with the same right one could choose a path that winds around the branching point zero multiple times, leading to multiple jumps. This is what @PaulSinclair commented, the integral is not well-defined, as it depends critically on the path chosen (and also the root branch selection).

You get less problems if you restrict $A,B$ to a convex set that does not contain the branching point/singularity $0$, and best also not the negative half-axis, the jump location of the main branch of the roots. For instance the half-plane of positive real parts fits these conditions.