Project-declaredLean 4.32.0 · mathlib@81a5d257c8e4

Integral deriv mul add const

integral_deriv_mul_add_const

analytic number theory prime numbers asymptotics euler maclaurin

Plain-language statement

An integration-by-parts identity for real- or complex-valued $f$ . If $a\le b$ , $f$ is differentiable on $[a,b]$ , and $f'$ is integrable there, then for every constant $c$ , $\int_a^b(t+c)f'(t)\,dt=(b+c)f(b)-(a+c)f(a)-\int_a^b f(t)\,dt.$

Exact Lean statement

lemma integral_deriv_mul_add_const (c : 𝕜) (hab : a ≤ b) (h_int : IntervalIntegrable (deriv f) volume a b)
    (hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t) :
    ∫ t in a..b, (t + c) * deriv f t = (b + c) * f b - (a + c) * f a - ∫ t in a..b, f t

Formal artifact

Lean source

Canonical source

Full Lean sourceLean 4

lemma integral_deriv_mul_add_const (c : 𝕜) (hab : a ≤ b) (h_int : IntervalIntegrable (deriv f) volume a b)    (hf_diff : ∀ t ∈ Set.Icc a b, DifferentiableAt ℝ f t) :    ∫ t in a..b, (t + c) * deriv f t = (b + c) * f b - (a + c) * f a - ∫ t in a..b, f t := by  rw [← Set.uIcc_of_le hab] at hf_diff  have : ∀ t ∈ [[a, b]], HasDerivAt (fun (t : ℝ) ↦ t + c) 1 t := by    intro t ht    simp only [hasDerivAt_add_const_iff]    convert! ContinuousLinearMap.hasDerivAt (RCLike.ofRealCLM (K := 𝕜)) using 1    simp  replace hf_diff := fun t ht ↦ (hf_diff t ht).hasDerivAt  rw [intervalIntegral.integral_mul_deriv_eq_deriv_mul this hf_diff (by simp) h_int]  simp

Project: Prime Number Theorem and More
License: Apache-2.0
Commit: a93551347dce
Source: PrimeNumberTheoremAnd/EulerMaclaurin.lean:28-39

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