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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|0VCTBJMEJF]] $\\int_a^\\infty f(x) \\, dx$ converges absolutely if $\\int_a^\\infty |f(x)| \\, dx$ converges.\n\n## Remarks\n\nAn absolutely convergent improper integral converges. The proof uses $0 \\le f + |f| \\le 2|f|$ and the [[comparison test|0FR1BDKK10]].\nThe converse fails. The [[Dirichlet integral|0GZNSHMG4Q]] $\\int_0^\\infty \\frac{\\sin x}{x} \\, dx$ converges, but not absolutely.\nA function on $[a, \\infty)$ whose improper integral converges absolutely is Lebesgue integrable there. The two integrals agree.\n\n## Explanations\n\n### L1 human\n\nAn improper integral converges absolutely when the area between the graph and the axis is finite, with all parts counted as positive.\nThis is the analogue of absolute convergence of series.\n",
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