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  "body": "\n## Statement\n\nLet $k \\ge 0$, let $a \\ne x$ be real numbers, and let $J$ be the closed interval with endpoints $a$ and $x$.\nLet $f$ be a real function on $J$ such that $f^{(k+1)}$ exists and is continuous on $J$.\nThen\n$$f(x) = P_k(x) + \\int_a^x \\frac{f^{(k+1)}(t)}{k!} (x - t)^k \\, dt,$$\nwhere $P_k$ is the [[Taylor polynomial|0HFSFVMWQB]] of degree $k$ of $f$ at $a$.\n\n## Remarks\n\nFor $k = 0$ it is the [[fundamental theorem of calculus|0NHGY0N0JB]], $f(x) = f(a) + \\int_a^x f'(t)\\, dt$. Repeated [[integration by parts|08Y1BBQTK9]] gives the general case.\nThe hypotheses can be weakened to absolute continuity of $f^{(k)}$.\nMathlib states it as `taylor_integral_remainder_of_absolutelyContinuous`.\n\n## Explanations\n\n### L1 human\n\nThe integral form gives the error exactly, as an average of the next derivative with weights.\nIt needs no unknown point $\\xi$.\n",
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  "msc": [
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  "name": "Taylor's theorem with the integral remainder",
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