In which cases and why can't we find any primitive?
If g is a Lebesgue-integrable function on some interval [a,b] and we define f as:
f(x) = \int_a^x g(t)dt
1. f is absolutely continuous
2. f is differentiable almost everywhere
6. it's derivative coincides almost everywhere with g(x)
So, if g is Lebesgue-integrable then, almost everywhere, f' = g (i.e it has an antiderivative).