theorem extracted_formal_statement_54.{u_1} {Ω : Type u_1} [inst : MeasurableSpace Ω] (μ : Measure Ω)
  [inst_1 : IsProbabilityMeasure μ] {n : ℕ} {X : Fin n → Ω → ℝ} (hXm : ∀ (i : Fin n), Measurable (X i))
  (hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1) (τ : ℝ) (i : Fin n) (h1 : Integrable (fun ω => 1 + τ * X i ω) μ)
  (h_int : Integrable (fun ω => X i ω ^ 2) μ) : Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ := sorry

State before:
Ω : Type u_1
inst : MeasurableSpace Ω
μ : Measure Ω
inst_1 : IsProbabilityMeasure μ
n : ℕ
X : Fin n → Ω → ℝ
hXm : ∀ (i : Fin n), Measurable (X i)
hX_bdd : ∀ (i : Fin n), ∀ᵐ (ω : Ω) ∂μ, |X i ω| ≤ 1
τ : ℝ
i : Fin n
h1 : Integrable (fun ω => 1 + τ * X i ω) μ
h_int : Integrable (fun ω => X i ω ^ 2) μ
⊢ Integrable (fun ω => X i ω ^ 2 * (rexp τ - 1 - τ)) μ

Tactic: apply Integrable.mul_const h_int (rexp τ - 1 - τ)

State after:

Relevant lemmas:
MeasureTheory.Integrable.mul_const : ∀ {α : Type u_1} {m : MeasurableSpace α} {μ : Measure α} {𝕜 : Type u_6}
  [inst : NormedRing 𝕜] {f : α → 𝕜}, Integrable f μ → ∀ (c : 𝕜), Integrable (fun x => f x * c) μ