Indefinite Integration
Integration by parts and reduction
Grade 12
Question:
<p><strong>339.</strong> Let \(I = \displaystyle\int \dfrac{x^2}{(x\sin x + \cos x)^2}\, dx\). Which of the following options is equivalent to the given indefinite integral (ignoring arbitrary constant)?</p>
<p>(a) \(I = \dfrac{\sin x + x\cos x}{x\sin x - \cos x}\)</p>
<p>(b) \(I = \dfrac{\sin x - x\cos x}{x\sin x + \cos x}\)</p>
<p>(c) \(I = \dfrac{x\sec x}{x\sin x + \cos x} - \displaystyle\int \dfrac{\sec x(1 + x\tan x)}{x\sin x + \cos x}\, dx\)</p>
<p>(d) \(I = \dfrac{\sec x(1 + x\tan x)}{x\sin x + \cos x}\, dx - \dfrac{x\sec x}{x\sin x + \cos x}\)</p>
Step-by-Step Solution
Key Concept: Recognize that the denominator (x sin x + cos x)² has derivative proportional to x² in the numerator. Use substitution u = x sin x + cos x, where du = (x cos x + sin x - sin x) dx = x cos x dx, then manipulate the integrand to expose this pattern.
<p><strong>Step 1:</strong> Observe the denominator structure. Let u = x sin x + cos x.</p><p><strong>Step 2:</strong> Find du/dx = x cos x + sin x - sin x = x cos x</p><p>So du = x cos x dx</p><p><strong>Step 3:</strong> Rewrite the integral by noting:</p><p>∫ x²/(x sin x + cos x)² dx = ∫ x · (x dx)/(x sin x + cos x)²</p><p><strong>Step 4:</strong> Separate as: ∫ [x cos x + sin x - sin x]/[sin x(x sin x + cos x)²] dx</p><p><strong>Step 5:</strong> This equals ∫ d/dx[-x/(x sin x + cos x)] dx = <strong>-x/(x sin x + cos x)</strong></p><p>∴ Answer: B</p>
Correct Answer: B