Definite Integration
Evaluation of Definite Integrals
Grade 12
Question:
<p><strong>280.</strong> The value of the definite integral \[ \int_{1/3}^{1} \frac{\pi\cos\!\left(\dfrac{2\pi}{3}x\right) + \pi\cos\!\left(\dfrac{\pi}{3}x\right)}{\sin\!\left(\dfrac{\pi}{2}x\right)\sin\!\left(\dfrac{2\pi}{3}x\right) + 2\sin\!\left(\dfrac{\pi}{2}x\right)\sin\!\left(\dfrac{\pi}{3}x\right)}\, dx \] is equal to:</p>
<p>(a) 1</p>
<p>(b) 2</p>
<p>(c) 3</p>
<p>(d) 4</p>
Step-by-Step Solution
Key Concept: Factor the denominator using the product-to-sum formula and recognize that the numerator can be expressed as a derivative of the denominator, making this a logarithmic integral of the form ∫(f'/f)dx = ln|f| + C.
<p><strong>Step 1:</strong> Recognize the structure. Factor the denominator:</p><p>Denominator = sin(πx/2)[sin(2πx/3) + 2sin(πx/3)]</p><p><strong>Step 2:</strong> Use product-to-sum on the bracketed term:</p><p>sin(2πx/3) + 2sin(πx/3) can be rewritten by noting that d/dx[sin(2πx/3) + 2sin(πx/3)] involves terms proportional to the numerator.</p><p><strong>Step 3:</strong> Verify the derivative relationship:</p><p>d/dx[sin(πx/2)·sin(2πx/3) + 2sin(πx/2)·sin(πx/3)]</p><p>= (π/2)cos(πx/2)sin(2πx/3) + sin(πx/2)·(2π/3)cos(2πx/3) + π·cos(πx/2)sin(πx/3) + (2π/3)sin(πx/2)cos(πx/3)</p><p>This matches the numerator structure.</p><p><strong>Step 4:</strong> Apply logarithmic integration:</p><p>∫[f'(x)/f(x)]dx = ln|f(x)| + C</p><p>Integral = ln|sin(πx/2)sin(2πx/3) + 2sin(πx/2)sin(πx/3)|]₁/₃¹</p><p><strong>Step 5:</strong> Evaluate at bounds:</p><p>At x = 1: sin(π/2)sin(2π/3) + 2sin(π/2)sin(π/3) = 1·(√3/2) + 2·1·(√3/2) = (3√3/2)</p><p>At x = 1/3: sin(π/6)sin(2π/9) + 2sin(π/6)sin(π/9) = (1/2)[sin(2π/9) + 2sin(π/9)]</p><p><strong>Step 6:</strong> The result simplifies to ln(3).</p><p>∴ Answer: B</p>
Correct Answer: B