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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $f \\colon [a, b] \\to \\mathbb{R}$ be continuous.\nThen the function $F(x) = \\int_a^x f(t) \\, dt$ is differentiable on $(a, b)$, with $F'(x) = f(x)$.\nIf $G$ is continuous on $[a, b]$ and $G' = f$ on $(a, b)$, then\n$$\\int_a^b f(x) \\, dx = G(b) - G(a).$$\n\n## Remarks\n\nThe first part is the [[first fundamental theorem|063H5KJC69]]. The second part is the [[second fundamental theorem|0XYAFCN0PT]].\nTogether they say that differentiation and integration are inverse operations.\nThe theorem reduces the computation of integrals to the search for [[antiderivatives|0BW9N5CDSB]].\n\n## History\n\nJames Gregory published the first statement and proof of a geometric form of the theorem.\nIsaac Barrow proved a more general version. Isaac Newton and Gottfried Leibniz then built the calculus around it.\n\n## Explanations\n\n### L1 human\n\nThe fundamental theorem connects the two main ideas of calculus.\nThe derivative of an area function is the height of the graph, and an integral is a change in an antiderivative.\n",
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