Henstock–Kurzweil integral

Summary

The Henstock–Kurzweil integral, also called the gauge integral, is an integral of Riemann type. In its sums the allowed size of a subinterval may vary from point to point. This one change makes it integrate every derivative, so that $\int_a^b F' = F(b) - F(a)$ holds whenever $F$ is differentiable everywhere on $[a, b]$. It integrates every Lebesgue integrable function with the same value, and it gives the same integral as the Perron integral. Unlike the Lebesgue integral, it also integrates some functions whose absolute value is not integrable.