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  "body": "\n## Statement\n\nLet $a, b$ be real numbers of the same sign, both nonzero.\nThen\n$$\\int_a^b \\frac{dx}{x} = \\ln \\frac{b}{a}.$$\nIn particular, $\\int_1^x \\frac{dt}{t} = \\ln x$ for $x \u003e 0$.\n\n## Remarks\n\nThe function $\\ln |x|$ is an antiderivative of $1/x$ on each of $(0, \\infty)$ and $(-\\infty, 0)$, by the [[derivative of the logarithm|07X1QA6BRW]].\nSome texts use the second formula as the definition of the [[natural logarithm|0044CPHJBV]].\nThe integral over an interval that contains $0$ does not exist.\nMathlib states it as `integral_inv`, under the hypothesis $0 \\notin [a, b]$.\n\n## Explanations\n\n### L1 human\n\nThe power rule fails for $x^{-1}$, since it would divide by $0$.\nThe logarithm fills this gap: its derivative is $1/x$.\n",
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