theorem extracted_formal_statement_106.{u_1} {Ω : Type u_1} {m : MeasurableSpace Ω} {μ : Measure Ω}
  [inst : IsFiniteMeasure μ] [inst_1 : NeZero μ] (a b : ℝ) {X : Ω → ℝ} (hX : AEMeasurable X μ)
  (h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b) (t : ℝ) :
  (∫ (x : Ω), (fun ω => id (X ω)) x ∂μ.tilted fun ω => t * X ω) =
    (∫ (x : Ω), (fun ω => rexp (t * X ω) * id (X ω)) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ := sorry

State before:
case h
Ω : Type u_1
m : MeasurableSpace Ω
μ : Measure Ω
inst✝¹ : IsFiniteMeasure μ
inst✝ : NeZero μ
a b : ℝ
X : Ω → ℝ
hX : AEMeasurable X μ
h : ∀ᵐ (ω : Ω) ∂μ, X ω ∈ Set.Icc a b
g' : ℝ → ℝ := fun t => ∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω
g'' : ℝ → ℝ :=
  fun t => (∫ (x : Ω), (X ^ 2) x ∂μ.tilted fun ω => t * X ω) - (∫ (x : Ω), X x ∂μ.tilted fun ω => t * X ω) ^ 2
t✝ t : ℝ
⊢ (∫ (x : Ω), (fun ω => id (X ω)) x ∂μ.tilted fun ω => t * X ω) =
    (∫ (x : Ω), (fun ω => rexp (t * X ω) * id (X ω)) x ∂μ) / ∫ (x : Ω), (fun ω => rexp (t * X ω)) x ∂μ

Tactic: exact integral_tilted t id hX

State after:

Relevant lemmas:
MeasureTheory.integral_tilted : ∀ {α : Type u_1} {mα : MeasurableSpace α} {μ : Measure α} {E : Type u_2}
  [inst : NormedAddCommGroup E] [inst_2 : NormedSpace ℝ E] (f : α → ℝ) (g : α → E),
  ∫ (x : α), g x ∂μ.tilted f = ∫ (x : α), (rexp (f x) / ∫ (x : α), rexp (f x) ∂μ) • g x ∂μ