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  "body": "\n## Summary\n\nThe Jackson integral, or $q$-integral, is a [[q-analog|GFSDMJHHVA]] of the [[Riemann integral|0W29TBTY12]].\nIt replaces the integral over $[0, a]$ by a weighted sum of the values of the function at the geometric sequence of points $a, aq, aq^2, \\dots$.\nIt is the inverse of the [[q-derivative|PMJT3FZE0Y]] in the same way as an integral is the inverse of a derivative.\nMany identities for basic hypergeometric series, such as $q$-beta integrals, are stated with it.\n\n## Explanations\n\n### L1 human\n\nA Riemann sum cuts $[0, a]$ into pieces of equal length.\nThe Jackson integral instead cuts it at the points $a q^k$, so the pieces become shorter and shorter near $0$.\nOn each piece $[a q^{k+1}, a q^k]$ it takes the value at the right end times the length $(1 - q) a q^k$.\nFor example, $\\int_0^1 x \\, d_q x = \\frac{1}{1 + q}$, which tends to $\\frac12$ as $q \\to 1$.\n",
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