Definite Integration
Evaluation of Definite Integrals
Grade 12
Question:
<p>Find the value of the definite integral \[\left(\frac{1}{\pi}\int_0^{\pi/2} \frac{\cos^4 x + \sin x\cos^3 x + \sin^2 x\cos^2 x + \sin^3 x\cos x}{\sin^4 x + \cos^4 x + 2\sin x\cos^3 x + 2\sin^2 x\cos^2 x + 2\sin^3 x\cos x}\,dx\right)^{-1}\]</p>
Step-by-Step Solution
Key Concept: Recognize that the denominator factors as (sin²x + cos²x)² = 1, and the numerator simplifies by factoring out cos x from strategic terms, reducing to a rational function whose integral evaluates via substitution.
<p><strong>Step 1: Simplify the denominator</strong></p><p>Denominator = sin⁴x + cos⁴x + 2sin x cos³x + 2sin²x cos²x + 2sin³x cos x</p><p>= sin⁴x + cos⁴x + 2sin x cos x(cos²x + sin x cos x + sin²x + sin²x)</p><p>= (sin²x + cos²x)² = 1</p><p><strong>Step 2: Simplify the numerator</strong></p><p>Numerator = cos⁴x + sin x cos³x + sin²x cos²x + sin³x cos x</p><p>= cos x(cos³x + sin cos²x + sin²x cos x + sin³x)</p><p>= cos x[cos²x(cos x + sin x) + sin²x(cos x + sin x)]</p><p>= cos x(cos x + sin x)(cos²x + sin²x) = cos x(cos x + sin x)</p><p><strong>Step 3: Evaluate the integral</strong></p><p>∫₀^(π/2) [cos x(cos x + sin x)]/1 dx = ∫₀^(π/2) (cos²x + sin x cos x) dx</p><p>= [x/2 + sin(2x)/4]₀^(π/2) + [sin²x/2]₀^(π/2)</p><p>= π/4 + 1/2 = (π + 2)/4</p><p><strong>Step 4: Apply the reciprocal</strong></p><p>(1/π · (π + 2)/4)⁻¹ = 4π/(π + 2)</p><p>∴ Answer: <strong>4π/(π + 2)</strong></p>
Correct Answer: 4