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  "body": "\n## Definition\n\nLet $f \\colon [a, \\infty) \\to \\mathbb{R}$ be Riemann integrable on $[a, b]$ for each $b \u003e a$.\nThe improper integral of $f$ over $[a, \\infty)$ converges if the following [[limit|0Y12QYTJK3]] exists in $\\mathbb{R}$:\n$$\\int_a^\\infty f(x) \\, dx = \\lim_{b \\to \\infty} \\int_a^b f(x) \\, dx.$$\nSimilarly, let $f \\colon (a, b] \\to \\mathbb{R}$ be Riemann integrable on $[c, b]$ for each $c \\in (a, b)$.\nThen $\\int_a^b f(x) \\, dx = \\lim_{c \\to a^+} \\int_c^b f(x) \\, dx$, if this [[one-sided limit|0909PWZR62]] exists.\n\n## Remarks\n\nIf the limit does not exist, the improper integral diverges.\nFor an integral over $(-\\infty, \\infty)$, or with several bad points, we split the interval. Each piece must converge separately.\nFor example, $\\int_1^\\infty \\frac{dx}{x^2} = 1$, and $\\int_0^1 \\frac{dx}{\\sqrt{x}} = 2$. But $\\int_1^\\infty \\frac{dx}{x}$ diverges.\n\n## Explanations\n\n### L1 human\n\nThe Riemann integral needs a bounded interval and a bounded function.\nAn improper integral removes these limits by a limit of ordinary integrals.\n",
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