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  "body": "\n## Summary\n\nThe Hadamard finite part gives a value to an integral that diverges because the integrand has a non-integrable singularity, such as $\\int_{-1}^{1} \\frac{\\varphi(t)}{t^2}\\, dt$.\nOne cuts out a small neighbourhood of the singularity, computes the integral, and drops the terms that blow up as the neighbourhood shrinks.\nIt extends the [[Cauchy principal value|0SYR10HD4R]], which handles poles of first order.\nFinite part integrals are the integrals in [[hypersingular integral equations|QVRP8FSAZM]].\n\n## Explanations\n\n### L1 human\n\nThe integral of $1/t^2$ over $(\\varepsilon, 1)$ is $1/\\varepsilon - 1$, which grows without bound as $\\varepsilon \\to 0$.\nThe finite part throws away the growing piece $1/\\varepsilon$ and keeps $-1$.\nSo a positive function can have a negative finite part integral, and this is not a mistake.\nThe rule is useful because it agrees with differentiating a principal value integral under the integral sign.\n",
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