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  "body": "\n## Summary\n\nThe Henstock–Kurzweil integral, also called the gauge integral, is an integral of Riemann type.\nIn its sums the allowed size of a subinterval may vary from point to point.\nThis one change makes it integrate every derivative, so that $\\int_a^b F' = F(b) - F(a)$ holds whenever $F$ is differentiable everywhere on $[a, b]$.\nIt integrates every [[Lebesgue integrable|DWJF7ABPR1]] function with the same value, and it gives the same integral as the [[Perron integral|M077AXJACA]].\nUnlike the Lebesgue integral, it also integrates some functions whose absolute value is not integrable.\n\n## Explanations\n\n### L1 human\n\nIn a Riemann sum all pieces must be short, with one bound on their length.\nIn the Henstock–Kurzweil sum each point $t$ comes with its own allowed size $\\delta(t)$, and a piece tagged by $t$ must fit within distance $\\delta(t)$ of $t$.\nNear a point where the function misbehaves, one demands very short pieces; elsewhere longer ones are fine.\nFor example, the derivative of $x^2 \\sin(1/x^2)$ is not Lebesgue integrable on $[0, 1]$, but it is Henstock–Kurzweil integrable with integral $\\sin 1$.\n",
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