case h.e'_3
a b : ℝ
n : ℕ
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
a b : ℝ
n : ℕ
⊢ (∫ (x : ℝ) in a..b, sin x ^ 2 * cos x ^ 2) = (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32
convert integral_sin_pow_even_mul_cos_pow_even 1 1 using 1
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
case h.e'_3
a b : ℝ
n : ℕ
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
have h1 : ∀ c : ℝ, (↑1 - c) / ↑2 * ((↑1 + c) / ↑2) = (↑1 - c ^ 2) / 4 := fun c => by ring
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
have h2 : Continuous fun x => cos (2 * x) ^ 2 := by continuity
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
have h3 : ∀ x, cos x * sin x = sin (2 * x) / 2 := by intro; rw [sin_two_mul]; ring
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
have h4 : ∀ d : ℝ, 2 * (2 * d) = 4 * d := fun d => by ring
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 = ∫ (x : ℝ) in a..b, (1 - cos (2 * x) ^ 2) / 4
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    ∫ (x : ℝ) in a..b, ((1 - cos (2 * x)) / 2) ^ 1 * ((1 + cos (2 * x)) / 2) ^ 1
simp only [pow_one, h1]
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 = (b - a - ∫ (x : ℝ) in a..b, cos (2 * x) ^ 2) / 4

case h.e'_3.hf
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ IntervalIntegrable (fun x => 1) MeasureTheory.volume a b

case h.e'_3.hg
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ IntervalIntegrable (fun x => cos (2 * x) ^ 2) MeasureTheory.volume a b
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 = ∫ (x : ℝ) in a..b, (1 - cos (2 * x) ^ 2) / 4
rw [integral_div, integral_sub, integral_one]
no goals
a b : ℝ
n : ℕ
c : ℝ
⊢ (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
ring
no goals
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
⊢ Continuous fun x => cos (2 * x) ^ 2
continuity
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
x✝ : ℝ
⊢ cos x✝ * sin x✝ = sin (2 * x✝) / 2
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
⊢ ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
intro
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
x✝ : ℝ
⊢ cos x✝ * sin x✝ = 2 * sin x✝ * cos x✝ / 2
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
x✝ : ℝ
⊢ cos x✝ * sin x✝ = sin (2 * x✝) / 2
rw [sin_two_mul]
no goals
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
x✝ : ℝ
⊢ cos x✝ * sin x✝ = 2 * sin x✝ * cos x✝ / 2
ring
no goals
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
d : ℝ
⊢ 2 * (2 * d) = 4 * d
ring
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    (b - a - 2⁻¹ * ((sin (4 * b) / 2 - sin (4 * a) / 2 + 2 * b - 2 * a) / 2)) / 4
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 = (b - a - ∫ (x : ℝ) in a..b, cos (2 * x) ^ 2) / 4
simp [integral_comp_mul_left (fun x => cos x ^ 2), h3, h4]
no goals
case h.e'_3
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ (b - a) / 8 - (sin (4 * b) - sin (4 * a)) / 32 =
    (b - a - 2⁻¹ * ((sin (4 * b) / 2 - sin (4 * a) / 2 + 2 * b - 2 * a) / 2)) / 4
ring
no goals
case h.e'_3.hf
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ IntervalIntegrable (fun x => 1) MeasureTheory.volume a b
exact intervalIntegrable_const
no goals
case h.e'_3.hg
a b : ℝ
n : ℕ
h1 : ∀ (c : ℝ), (1 - c) / 2 * ((1 + c) / 2) = (1 - c ^ 2) / 4
h2 : Continuous fun x => cos (2 * x) ^ 2
h3 : ∀ (x : ℝ), cos x * sin x = sin (2 * x) / 2
h4 : ∀ (d : ℝ), 2 * (2 * d) = 4 * d
⊢ IntervalIntegrable (fun x => cos (2 * x) ^ 2) MeasureTheory.volume a b
exact h2.intervalIntegrable a b