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  "body": "\n## Statement\n\nLet $a \u003c b$ and let $u, v \\colon [a, b] \\to \\mathbb{R}$ be [[continuously differentiable|06SKCHDQT5]].\nThen\n$$\\int_a^b u(x) v'(x) \\, dx = u(b) v(b) - u(a) v(a) - \\int_a^b u'(x) v(x) \\, dx.$$\n\n## Remarks\n\nBy the [[product rule|0V9A43H7WV]], $uv$ is an antiderivative of $u v' + u' v$. Apply the [[second fundamental theorem|0XYAFCN0PT]].\nFor indefinite integrals, the formula is $\\int u \\, dv = u v - \\int v \\, du$.\nFor example, $\\int_0^1 x e^x \\, dx = e - (e - 1) = 1$.\nMathlib states it as `intervalIntegral.integral_mul_deriv_eq_deriv_mul`.\n\n## Explanations\n\n### L1 human\n\nIntegration by parts is the product rule read backwards.\nIt moves the derivative from one factor to the other, at the cost of a boundary term.\n",
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