integral_comp_mul_deriv — Mathlib

English

In the vector-valued IBP chain, replacing smul by inner product yields the same total derivative identity, i.e., the boundary terms minus the interior integral of the derivatives.

Русский

Для векторного IBP сохраняется та же структура: границы минус внутренний интеграл производных.

LaTeX

$$$\\displaystyle \\int_a^b u(x) \\cdot v'(x) \\,dx = u(b) \\cdot v(b) - u(a) \\cdot v(a) - \\int_a^b u'(x) \\cdot v(x) \\,dx$$$

Lean4

/-- Change of variables, most common version. If `f` has continuous derivative `f'` on `[a, b]`,
and `g` is continuous, then we can substitute `u = f x` to get
`∫ x in a..b, (g ∘ f) x * f' x = ∫ u in f a..f b, g u`.
-/
theorem integral_comp_mul_deriv {f f' g : ℝ → ℝ} (h : ∀ x ∈ uIcc a b, HasDerivAt f (f' x) x)
    (h' : ContinuousOn f' (uIcc a b)) (hg : Continuous g) : (∫ x in a..b, (g ∘ f) x * f' x) = ∫ x in f a..f b, g x :=
  integral_comp_mul_deriv' h h' hg.continuousOn

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