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  "body": "\n## Statement\n\nLet $a \u003c b$, and let $F \\colon [a, b] \\to \\mathbb{R}$ be continuous on $[a, b]$ and differentiable on $(a, b)$.\nAssume that $F'$ extends to a [[Riemann integrable|0N1JQXZXHC]] function $f$ on $[a, b]$.\nThen\n$$\\int_a^b f(x) \\, dx = F(b) - F(a).$$\n\n## Remarks\n\nFor a partition $P$, the [[mean value theorem|0BARQY52N5]] gives tags $t_i$ with $F(x_i) - F(x_{i-1}) = f(t_i)(x_i - x_{i-1})$. So $F(b) - F(a)$ is a [[Riemann sum|0HBY48G8G1]] of $f$ for each partition.\nOne writes $F(b) - F(a) = \\big[ F(x) \\big]_a^b$.\nThe hypothesis that $f$ is integrable is needed. Some differentiable functions have derivatives that are not Riemann integrable.\nMathlib states it as `intervalIntegral.integral_eq_sub_of_hasDeriv_right_of_le`.\n\n## Explanations\n\n### L1 human\n\nTo integrate $f$, find a function $F$ with derivative $f$.\nThe integral is the change of $F$ from $a$ to $b$.\n",
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