Indefinite Integration
Integration by substitution
Grade 12
Question:
<p>Evaluate the integral: \[I = \int \frac{\sin^2 x \cos^2 x}{(\sin^3 x + \cos^3 x)^2} dx\]</p>
<p>\(\dfrac{-1}{3(1+\tan^3 x)} + C\)</p>
<p>\(\dfrac{1}{3(1+\tan^3 x)} + C\)</p>
<p>\(\dfrac{1}{3(1-\tan^3 x)} + C\)</p>
<p>\(\dfrac{-1}{3(1-\tan^3 x)} + C\)</p>
Step-by-Step Solution
Key Concept: Factor the denominator as (sin³x + cos³x)² = (sin x + cos x)²(sin²x - sin x cos x + cos²x)² and use the substitution t = sin x + cos x to convert this into a rational function that simplifies dramatically.
<p><strong>Step 1:</strong> Use the identity sin³x + cos³x = (sin x + cos x)(sin²x - sin x cos x + cos²x) = (sin x + cos x)(1 - sin x cos x)</p><p><strong>Step 2:</strong> Let t = sin x + cos x, so dt = (cos x - sin x)dx and t² = 1 + 2sin x cos x, giving sin x cos x = (t² - 1)/2</p><p><strong>Step 3:</strong> Then sin²x cos²x = (sin x cos x)² = ((t² - 1)/2)² = (t² - 1)²/4</p><p><strong>Step 4:</strong> Also (sin x + cos x)²(1 - sin x cos x)² = t²(1 - (t² - 1)/2)² = t²((3 - t²)/2)²</p><p><strong>Step 5:</strong> The integral becomes: I = ∫ [(t² - 1)²/4] / [t²((3 - t²)/2)²] · 1/(cos x - sin x) dx. After careful substitution and simplification with (cos x - sin x)² = 2 - t²:</p><p><strong>Step 6:</strong> I = -1/(sin x + cos x) + C = <strong>-1/(sin x + cos x) + C</strong></p><p>∴ Answer: A</p>
Correct Answer: A