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  "body": "\n## Definition\n\nLet $a \u003c b \u003c c$ be [[real numbers|08J2M1JYAG]].\nLet $f$ be a function on $[a, c] \\setminus \\{b\\}$ that is integrable on $[a, b - \\varepsilon]$ and on $[b + \\varepsilon, c]$ for each $\\varepsilon$ with $0 \u003c \\varepsilon \u003c \\min(b - a, c - b)$.\nIf the limit\n$$\\operatorname{p.v.} \\int_a^c f(x) \\, dx = \\lim_{\\varepsilon \\to 0^+} \\left( \\int_a^{b - \\varepsilon} f(x) \\, dx + \\int_{b + \\varepsilon}^{c} f(x) \\, dx \\right)$$\nexists, it is the Cauchy principal value of the integral of $f$ over $[a, c]$.\nLet $f$ be a function on $\\mathbb{R}$ that is integrable on each interval $[-R, R]$.\nIf the limit $\\operatorname{p.v.} \\int_{-\\infty}^{\\infty} f(x) \\, dx = \\lim_{R \\to \\infty} \\int_{-R}^{R} f(x) \\, dx$ exists, it is the Cauchy principal value of the integral of $f$ over $\\mathbb{R}$.\n\n## Remarks\n\nThe integral $\\int_{-1}^{1} \\frac{dx}{x}$ does not exist as an improper integral, but $\\operatorname{p.v.} \\int_{-1}^{1} \\frac{dx}{x} = 0$, because the two parts cancel.\nSimilarly, $\\operatorname{p.v.} \\int_{-\\infty}^{\\infty} x \\, dx = 0$, because $\\int_{-R}^{R} x \\, dx = 0$ for each $R$.\nWhen the improper integral exists, the principal value exists and has the same value.\nPrincipal values occur in the [[evaluation of real integrals by residues|0476QH9846]] when the integrand has a pole on the real axis.\n\n## Explanations\n\n### L1 human\n\nSometimes an integral is infinite on the left of a bad point and infinite on the right, with opposite signs.\nThe principal value cuts out a small symmetric interval around the bad point, so that the two sides can cancel.\nFor $\\frac{1}{x}$ on $[-1, 1]$, the areas on the two sides cancel exactly.\n",
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