Indefinite Integration
Integration by substitution
Grade 12
Question:
<p>If \(\int \frac{\sqrt{1-x^2}}{x^4} dx = A(x)(\sqrt{1-x^2})^m + C\), for a suitable chosen integer \(m\) and a function \(A(x)\), where \(C\) is a constant of integration, then \((A(x))^m\) equals:</p>
<p>\(\dfrac{1}{27x^6}\)</p>
<p>\(\dfrac{-1}{3x^3}\)</p>
<p>\(\dfrac{-1}{27x^9}\)</p>
<p>\(\dfrac{1}{9x^4}\)</p>
Step-by-Step Solution
Key Concept: Use substitution x = sin(θ) to convert the integrand into trigonometric form, then integrate and express the result as a power of (√(1-x²)) multiplied by a function A(x). The exponent m and function A(x) are determined by matching the integrated form with the given template.
<p><strong>Step 1:</strong> Use substitution x = sin(θ), so dx = cos(θ)dθ and √(1-x²) = cos(θ)</p><p>∫ (√(1-x²))/x⁴ dx = ∫ cos(θ)/sin⁴(θ) · cos(θ)dθ = ∫ cos²(θ)/sin⁴(θ) dθ</p><p><strong>Step 2:</strong> Rewrite as ∫ cot²(θ)csc²(θ)dθ. Let u = cot(θ), du = -csc²(θ)dθ</p><p>∫ cos²(θ)/sin⁴(θ) dθ = -∫ u² du = -u³/3 + C = -cot³(θ)/3 + C</p><p><strong>Step 3:</strong> Back-substitute. cot(θ) = cos(θ)/sin(θ) = √(1-x²)/x</p><p>-1/3 · (√(1-x²)/x)³ + C = -1/(3x³) · (√(1-x²))³ + C</p><p><strong>Step 4:</strong> Express in the form A(x)(√(1-x²))^m</p><p>A(x) = -1/(3x³) and m = 3</p><p><strong>Step 5:</strong> Calculate (A(x))^m = (-1/(3x³))³ = -1/(27x⁹)</p><p>∴ Answer: (A(x))³ = <strong>-1/(27x⁹)</strong></p>
Correct Answer: C