Is there a function f = g' but not Riemann Integrable

The second fundamental theorem of calculus states that:

If f is integrable in [a, b] and f = g' for some function g, then

$\displaystyle \int_a^b\! f(x) \, dx = g(b) - g(a)$

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so I was wondering if there is a function such that it is not Riemann integrable ( $\displaystyle \exists \epsilon, U(f,P) - L(f,P) \geq \epsilon$, or equivalent formulations), and f = g' for some function g.

Henstock-Kurzweil integral case

By any chance do you know Laurent, or any one else if that is also the case with the Henstock-Kurzweil integral (i.e. there's a function such that it is derivable, but it's derivative is not henstock-kurzweil integrable)?