Definite Integration
Reduction formula / Beta function
Grade 12
Question:
<p>If <span>\(I = \int_0^1 (1-x^4)^7\, dx\)</span>, then the value of <span>\(\dfrac{29\int_0^1(1-x^4)^7\,dx}{10\int_0^1(1-x^4)^6\,dx}\)</span> is:</p>
Step-by-Step Solution
Key Concept: Use the reduction formula for integrals of the form ∫₀¹(1-xⁿ)ᵐ dx, which relates consecutive powers through the beta function or by deriving a recurrence relation via integration by parts.
<p><strong>Step 1:</strong> Let Iₙ = ∫₀¹(1-x⁴)ⁿ dx. We need to find the ratio 29I₇/(10I₆).</p><p><strong>Step 2:</strong> Derive the reduction formula by integration by parts. Let u = (1-x⁴)ⁿ, dv = dx:</p><p>Iₙ = [x(1-x⁴)ⁿ]₀¹ + 4n∫₀¹ x⁴(1-x⁴)ⁿ⁻¹ dx = 4n∫₀¹ x⁴(1-x⁴)ⁿ⁻¹ dx</p><p><strong>Step 3:</strong> Write x⁴(1-x⁴)ⁿ⁻¹ = (1-x⁴)ⁿ⁻¹ - (1-x⁴)ⁿ:</p><p>Iₙ = 4n[Iₙ₋₁ - Iₙ]</p><p>Iₙ(1 + 4n) = 4nIₙ₋₁</p><p>Iₙ = (4n/(4n+1))Iₙ₋₁</p><p><strong>Step 4:</strong> Apply the relation I₇ = (28/29)I₆:</p><p>29I₇/(10I₆) = 29 · (28/29)I₆/(10I₆) = 28/10 = 14/5 = 2.80</p><p>∴ Answer: <strong>2.80</strong></p>
Correct Answer: 2.80