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L3 Lab 3

a b : ℝ n : ℕ f : ℝ → ℝ μ ν : Measure ℝ inst✝ : IsLocallyFiniteMeasure μ c d r : ℝ h : -1 < r this : ∀ (c : ℝ), IntervalIntegrable (fun x => x ^ r) volume 0 c ⊢ IntervalIntegrable (fun x => x ^ r) volume a b

/- Copyright (c) 2021 Benjamin Davidson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Benjamin Davidson -/ import Mathlib.MeasureTheory.Integral.FundThmCalculus import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv import Mathlib.Analysis.SpecialFunctions.NonIntegrable #align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011cafb4a5bc695875d186e245d6b3df03bf6c40" /-! # Integration of specific interval integrals This file contains proofs of the integrals of various specific functions. This includes: * Integrals of simple functions, such as `id`, `pow`, `inv`, `exp`, `log` * Integrals of some trigonometric functions, such as `sin`, `cos`, `1 / (1 + x^2)` * The integral of `cos x ^ 2 - sin x ^ 2` * Reduction formulae for the integrals of `sin x ^ n` and `cos x ^ n` for `n ≥ 2` * The computation of `∫ x in 0..π, sin x ^ n` as a product for even and odd `n` (used in proving the Wallis product for pi) * Integrals of the form `sin x ^ m * cos x ^ n` With these lemmas, many simple integrals can be computed by `simp` or `norm_num`. See `test/integration.lean` for specific examples. This file also contains some facts about the interval integrability of specific functions. This file is still being developed. ## Tags integrate, integration, integrable, integrability -/ open Real Nat Set Finset open scoped Real BigOperators Interval variable {a b : ℝ} (n : ℕ) namespace intervalIntegral open MeasureTheory variable {f : ℝ → ℝ} {μ ν : Measure ℝ} [IsLocallyFiniteMeasure μ] (c d : ℝ) /-! ### Interval integrability -/ @[simp] theorem intervalIntegrable_pow : IntervalIntegrable (fun x => x ^ n) μ a b := (continuous_pow n).intervalIntegrable a b #align interval_integral.interval_integrable_pow intervalIntegral.intervalIntegrable_pow theorem intervalIntegrable_zpow {n : ℤ} (h : 0 ≤ n ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ n) μ a b := (continuousOn_id.zpow₀ n fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_zpow intervalIntegral.intervalIntegrable_zpow /-- See `intervalIntegrable_rpow'` for a version with a weaker hypothesis on `r`, but assuming the measure is volume. -/ theorem intervalIntegrable_rpow {r : ℝ} (h : 0 ≤ r ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ r) μ a b := (continuousOn_id.rpow_const fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_rpow intervalIntegral.intervalIntegrable_rpow /-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by

exact IntervalIntegrable.trans (this a).symm (this b)

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by

Mathlib.Analysis.SpecialFunctions.Integrals.70_0.61KIRGZyZnxsI7s

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b

Mathlib_Analysis_SpecialFunctions_Integrals

a b : ℝ n : ℕ f : ℝ → ℝ μ ν : Measure ℝ inst✝ : IsLocallyFiniteMeasure μ c d r : ℝ h : -1 < r ⊢ ∀ (c : ℝ), IntervalIntegrable (fun x => x ^ r) volume 0 c

/- Copyright (c) 2021 Benjamin Davidson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Benjamin Davidson -/ import Mathlib.MeasureTheory.Integral.FundThmCalculus import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv import Mathlib.Analysis.SpecialFunctions.NonIntegrable #align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011cafb4a5bc695875d186e245d6b3df03bf6c40" /-! # Integration of specific interval integrals This file contains proofs of the integrals of various specific functions. This includes: * Integrals of simple functions, such as `id`, `pow`, `inv`, `exp`, `log` * Integrals of some trigonometric functions, such as `sin`, `cos`, `1 / (1 + x^2)` * The integral of `cos x ^ 2 - sin x ^ 2` * Reduction formulae for the integrals of `sin x ^ n` and `cos x ^ n` for `n ≥ 2` * The computation of `∫ x in 0..π, sin x ^ n` as a product for even and odd `n` (used in proving the Wallis product for pi) * Integrals of the form `sin x ^ m * cos x ^ n` With these lemmas, many simple integrals can be computed by `simp` or `norm_num`. See `test/integration.lean` for specific examples. This file also contains some facts about the interval integrability of specific functions. This file is still being developed. ## Tags integrate, integration, integrable, integrability -/ open Real Nat Set Finset open scoped Real BigOperators Interval variable {a b : ℝ} (n : ℕ) namespace intervalIntegral open MeasureTheory variable {f : ℝ → ℝ} {μ ν : Measure ℝ} [IsLocallyFiniteMeasure μ] (c d : ℝ) /-! ### Interval integrability -/ @[simp] theorem intervalIntegrable_pow : IntervalIntegrable (fun x => x ^ n) μ a b := (continuous_pow n).intervalIntegrable a b #align interval_integral.interval_integrable_pow intervalIntegral.intervalIntegrable_pow theorem intervalIntegrable_zpow {n : ℤ} (h : 0 ≤ n ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ n) μ a b := (continuousOn_id.zpow₀ n fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_zpow intervalIntegral.intervalIntegrable_zpow /-- See `intervalIntegrable_rpow'` for a version with a weaker hypothesis on `r`, but assuming the measure is volume. -/ theorem intervalIntegrable_rpow {r : ℝ} (h : 0 ≤ r ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ r) μ a b := (continuousOn_id.rpow_const fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_rpow intervalIntegral.intervalIntegrable_rpow /-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by exact IntervalIntegrable.trans (this a).symm (this b)

have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by intro c hc rw [intervalIntegrable_iff, uIoc_of_le hc] have hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x : ℝ => x ^ (r + 1) / (r + 1)) (x ^ r) x := by intro x hx convert (Real.hasDerivAt_rpow_const (p := r + 1) (Or.inl hx.1.ne')).div_const (r + 1) using 1 field_simp [(by linarith : r + 1 ≠ 0)]; ring apply integrableOn_deriv_of_nonneg _ hderiv · intro x hx; apply rpow_nonneg_of_nonneg hx.1.le · refine' (continuousOn_id.rpow_const _).div_const _; intro x _; right; linarith

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by exact IntervalIntegrable.trans (this a).symm (this b)

Mathlib.Analysis.SpecialFunctions.Integrals.70_0.61KIRGZyZnxsI7s

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b

Mathlib_Analysis_SpecialFunctions_Integrals

a b : ℝ n : ℕ f : ℝ → ℝ μ ν : Measure ℝ inst✝ : IsLocallyFiniteMeasure μ c d r : ℝ h : -1 < r ⊢ ∀ (c : ℝ), 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c

/- Copyright (c) 2021 Benjamin Davidson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Benjamin Davidson -/ import Mathlib.MeasureTheory.Integral.FundThmCalculus import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv import Mathlib.Analysis.SpecialFunctions.NonIntegrable #align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011cafb4a5bc695875d186e245d6b3df03bf6c40" /-! # Integration of specific interval integrals This file contains proofs of the integrals of various specific functions. This includes: * Integrals of simple functions, such as `id`, `pow`, `inv`, `exp`, `log` * Integrals of some trigonometric functions, such as `sin`, `cos`, `1 / (1 + x^2)` * The integral of `cos x ^ 2 - sin x ^ 2` * Reduction formulae for the integrals of `sin x ^ n` and `cos x ^ n` for `n ≥ 2` * The computation of `∫ x in 0..π, sin x ^ n` as a product for even and odd `n` (used in proving the Wallis product for pi) * Integrals of the form `sin x ^ m * cos x ^ n` With these lemmas, many simple integrals can be computed by `simp` or `norm_num`. See `test/integration.lean` for specific examples. This file also contains some facts about the interval integrability of specific functions. This file is still being developed. ## Tags integrate, integration, integrable, integrability -/ open Real Nat Set Finset open scoped Real BigOperators Interval variable {a b : ℝ} (n : ℕ) namespace intervalIntegral open MeasureTheory variable {f : ℝ → ℝ} {μ ν : Measure ℝ} [IsLocallyFiniteMeasure μ] (c d : ℝ) /-! ### Interval integrability -/ @[simp] theorem intervalIntegrable_pow : IntervalIntegrable (fun x => x ^ n) μ a b := (continuous_pow n).intervalIntegrable a b #align interval_integral.interval_integrable_pow intervalIntegral.intervalIntegrable_pow theorem intervalIntegrable_zpow {n : ℤ} (h : 0 ≤ n ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ n) μ a b := (continuousOn_id.zpow₀ n fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_zpow intervalIntegral.intervalIntegrable_zpow /-- See `intervalIntegrable_rpow'` for a version with a weaker hypothesis on `r`, but assuming the measure is volume. -/ theorem intervalIntegrable_rpow {r : ℝ} (h : 0 ≤ r ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ r) μ a b := (continuousOn_id.rpow_const fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_rpow intervalIntegral.intervalIntegrable_rpow /-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by exact IntervalIntegrable.trans (this a).symm (this b) have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by

intro c hc

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by exact IntervalIntegrable.trans (this a).symm (this b) have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by

Mathlib.Analysis.SpecialFunctions.Integrals.70_0.61KIRGZyZnxsI7s

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b

Mathlib_Analysis_SpecialFunctions_Integrals

a b : ℝ n : ℕ f : ℝ → ℝ μ ν : Measure ℝ inst✝ : IsLocallyFiniteMeasure μ c✝ d r : ℝ h : -1 < r c : ℝ hc : 0 ≤ c ⊢ IntervalIntegrable (fun x => x ^ r) volume 0 c

/- Copyright (c) 2021 Benjamin Davidson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Benjamin Davidson -/ import Mathlib.MeasureTheory.Integral.FundThmCalculus import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv import Mathlib.Analysis.SpecialFunctions.NonIntegrable #align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011cafb4a5bc695875d186e245d6b3df03bf6c40" /-! # Integration of specific interval integrals This file contains proofs of the integrals of various specific functions. This includes: * Integrals of simple functions, such as `id`, `pow`, `inv`, `exp`, `log` * Integrals of some trigonometric functions, such as `sin`, `cos`, `1 / (1 + x^2)` * The integral of `cos x ^ 2 - sin x ^ 2` * Reduction formulae for the integrals of `sin x ^ n` and `cos x ^ n` for `n ≥ 2` * The computation of `∫ x in 0..π, sin x ^ n` as a product for even and odd `n` (used in proving the Wallis product for pi) * Integrals of the form `sin x ^ m * cos x ^ n` With these lemmas, many simple integrals can be computed by `simp` or `norm_num`. See `test/integration.lean` for specific examples. This file also contains some facts about the interval integrability of specific functions. This file is still being developed. ## Tags integrate, integration, integrable, integrability -/ open Real Nat Set Finset open scoped Real BigOperators Interval variable {a b : ℝ} (n : ℕ) namespace intervalIntegral open MeasureTheory variable {f : ℝ → ℝ} {μ ν : Measure ℝ} [IsLocallyFiniteMeasure μ] (c d : ℝ) /-! ### Interval integrability -/ @[simp] theorem intervalIntegrable_pow : IntervalIntegrable (fun x => x ^ n) μ a b := (continuous_pow n).intervalIntegrable a b #align interval_integral.interval_integrable_pow intervalIntegral.intervalIntegrable_pow theorem intervalIntegrable_zpow {n : ℤ} (h : 0 ≤ n ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ n) μ a b := (continuousOn_id.zpow₀ n fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_zpow intervalIntegral.intervalIntegrable_zpow /-- See `intervalIntegrable_rpow'` for a version with a weaker hypothesis on `r`, but assuming the measure is volume. -/ theorem intervalIntegrable_rpow {r : ℝ} (h : 0 ≤ r ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ r) μ a b := (continuousOn_id.rpow_const fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_rpow intervalIntegral.intervalIntegrable_rpow /-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by exact IntervalIntegrable.trans (this a).symm (this b) have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by intro c hc

rw [intervalIntegrable_iff, uIoc_of_le hc]

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by exact IntervalIntegrable.trans (this a).symm (this b) have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by intro c hc

Mathlib.Analysis.SpecialFunctions.Integrals.70_0.61KIRGZyZnxsI7s

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b

Mathlib_Analysis_SpecialFunctions_Integrals

a b : ℝ n : ℕ f : ℝ → ℝ μ ν : Measure ℝ inst✝ : IsLocallyFiniteMeasure μ c✝ d r : ℝ h : -1 < r c : ℝ hc : 0 ≤ c ⊢ IntegrableOn (fun x => x ^ r) (Set.Ioc 0 c)

/- Copyright (c) 2021 Benjamin Davidson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Benjamin Davidson -/ import Mathlib.MeasureTheory.Integral.FundThmCalculus import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv import Mathlib.Analysis.SpecialFunctions.NonIntegrable #align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011cafb4a5bc695875d186e245d6b3df03bf6c40" /-! # Integration of specific interval integrals This file contains proofs of the integrals of various specific functions. This includes: * Integrals of simple functions, such as `id`, `pow`, `inv`, `exp`, `log` * Integrals of some trigonometric functions, such as `sin`, `cos`, `1 / (1 + x^2)` * The integral of `cos x ^ 2 - sin x ^ 2` * Reduction formulae for the integrals of `sin x ^ n` and `cos x ^ n` for `n ≥ 2` * The computation of `∫ x in 0..π, sin x ^ n` as a product for even and odd `n` (used in proving the Wallis product for pi) * Integrals of the form `sin x ^ m * cos x ^ n` With these lemmas, many simple integrals can be computed by `simp` or `norm_num`. See `test/integration.lean` for specific examples. This file also contains some facts about the interval integrability of specific functions. This file is still being developed. ## Tags integrate, integration, integrable, integrability -/ open Real Nat Set Finset open scoped Real BigOperators Interval variable {a b : ℝ} (n : ℕ) namespace intervalIntegral open MeasureTheory variable {f : ℝ → ℝ} {μ ν : Measure ℝ} [IsLocallyFiniteMeasure μ] (c d : ℝ) /-! ### Interval integrability -/ @[simp] theorem intervalIntegrable_pow : IntervalIntegrable (fun x => x ^ n) μ a b := (continuous_pow n).intervalIntegrable a b #align interval_integral.interval_integrable_pow intervalIntegral.intervalIntegrable_pow theorem intervalIntegrable_zpow {n : ℤ} (h : 0 ≤ n ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ n) μ a b := (continuousOn_id.zpow₀ n fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_zpow intervalIntegral.intervalIntegrable_zpow /-- See `intervalIntegrable_rpow'` for a version with a weaker hypothesis on `r`, but assuming the measure is volume. -/ theorem intervalIntegrable_rpow {r : ℝ} (h : 0 ≤ r ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ r) μ a b := (continuousOn_id.rpow_const fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_rpow intervalIntegral.intervalIntegrable_rpow /-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by exact IntervalIntegrable.trans (this a).symm (this b) have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by intro c hc rw [intervalIntegrable_iff, uIoc_of_le hc]

have hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x : ℝ => x ^ (r + 1) / (r + 1)) (x ^ r) x := by intro x hx convert (Real.hasDerivAt_rpow_const (p := r + 1) (Or.inl hx.1.ne')).div_const (r + 1) using 1 field_simp [(by linarith : r + 1 ≠ 0)]; ring

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by exact IntervalIntegrable.trans (this a).symm (this b) have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by intro c hc rw [intervalIntegrable_iff, uIoc_of_le hc]

Mathlib.Analysis.SpecialFunctions.Integrals.70_0.61KIRGZyZnxsI7s

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b

Mathlib_Analysis_SpecialFunctions_Integrals

a b : ℝ n : ℕ f : ℝ → ℝ μ ν : Measure ℝ inst✝ : IsLocallyFiniteMeasure μ c✝ d r : ℝ h : -1 < r c : ℝ hc : 0 ≤ c ⊢ ∀ x ∈ Set.Ioo 0 c, HasDerivAt (fun x => x ^ (r + 1) / (r + 1)) (x ^ r) x

/- Copyright (c) 2021 Benjamin Davidson. All rights reserved. Released under Apache 2.0 license as described in the file LICENSE. Authors: Benjamin Davidson -/ import Mathlib.MeasureTheory.Integral.FundThmCalculus import Mathlib.Analysis.SpecialFunctions.Trigonometric.ArctanDeriv import Mathlib.Analysis.SpecialFunctions.NonIntegrable #align_import analysis.special_functions.integrals from "leanprover-community/mathlib"@"011cafb4a5bc695875d186e245d6b3df03bf6c40" /-! # Integration of specific interval integrals This file contains proofs of the integrals of various specific functions. This includes: * Integrals of simple functions, such as `id`, `pow`, `inv`, `exp`, `log` * Integrals of some trigonometric functions, such as `sin`, `cos`, `1 / (1 + x^2)` * The integral of `cos x ^ 2 - sin x ^ 2` * Reduction formulae for the integrals of `sin x ^ n` and `cos x ^ n` for `n ≥ 2` * The computation of `∫ x in 0..π, sin x ^ n` as a product for even and odd `n` (used in proving the Wallis product for pi) * Integrals of the form `sin x ^ m * cos x ^ n` With these lemmas, many simple integrals can be computed by `simp` or `norm_num`. See `test/integration.lean` for specific examples. This file also contains some facts about the interval integrability of specific functions. This file is still being developed. ## Tags integrate, integration, integrable, integrability -/ open Real Nat Set Finset open scoped Real BigOperators Interval variable {a b : ℝ} (n : ℕ) namespace intervalIntegral open MeasureTheory variable {f : ℝ → ℝ} {μ ν : Measure ℝ} [IsLocallyFiniteMeasure μ] (c d : ℝ) /-! ### Interval integrability -/ @[simp] theorem intervalIntegrable_pow : IntervalIntegrable (fun x => x ^ n) μ a b := (continuous_pow n).intervalIntegrable a b #align interval_integral.interval_integrable_pow intervalIntegral.intervalIntegrable_pow theorem intervalIntegrable_zpow {n : ℤ} (h : 0 ≤ n ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ n) μ a b := (continuousOn_id.zpow₀ n fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_zpow intervalIntegral.intervalIntegrable_zpow /-- See `intervalIntegrable_rpow'` for a version with a weaker hypothesis on `r`, but assuming the measure is volume. -/ theorem intervalIntegrable_rpow {r : ℝ} (h : 0 ≤ r ∨ (0 : ℝ) ∉ [[a, b]]) : IntervalIntegrable (fun x => x ^ r) μ a b := (continuousOn_id.rpow_const fun _ hx => h.symm.imp (ne_of_mem_of_not_mem hx) id).intervalIntegrable #align interval_integral.interval_integrable_rpow intervalIntegral.intervalIntegrable_rpow /-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by exact IntervalIntegrable.trans (this a).symm (this b) have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by intro c hc rw [intervalIntegrable_iff, uIoc_of_le hc] have hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x : ℝ => x ^ (r + 1) / (r + 1)) (x ^ r) x := by

intro x hx

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b := by suffices ∀ c : ℝ, IntervalIntegrable (fun x => x ^ r) volume 0 c by exact IntervalIntegrable.trans (this a).symm (this b) have : ∀ c : ℝ, 0 ≤ c → IntervalIntegrable (fun x => x ^ r) volume 0 c := by intro c hc rw [intervalIntegrable_iff, uIoc_of_le hc] have hderiv : ∀ x ∈ Ioo 0 c, HasDerivAt (fun x : ℝ => x ^ (r + 1) / (r + 1)) (x ^ r) x := by

Mathlib.Analysis.SpecialFunctions.Integrals.70_0.61KIRGZyZnxsI7s

/-- See `intervalIntegrable_rpow` for a version applying to any locally finite measure, but with a stronger hypothesis on `r`. -/ theorem intervalIntegrable_rpow' {r : ℝ} (h : -1 < r) : IntervalIntegrable (fun x => x ^ r) volume a b

Mathlib_Analysis_SpecialFunctions_Integrals