Definite Integration
Evaluation of Definite Integrals
Grade 12
Question:
<p>The integral \(\displaystyle\int_0^{1/2}\dfrac{\ln(1+2x)}{1+4x^2}\,dx\) equals:</p>
<p>\(\dfrac{\pi}{4}\ln 2\)</p>
<p>\(\dfrac{\pi}{8}\ln 2\)</p>
<p>\(\dfrac{\pi}{16}\ln 2\)</p>
<p>\(\dfrac{\pi}{32}\ln 2\)</p>
Step-by-Step Solution
Key Concept: Use Feynman's trick by parameterizing the integrand with a parameter that allows differentiation under the integral sign, then exploit symmetry or substitution to reduce to a standard form.
<p><strong>Step 1:</strong> Define parameter: Let I(α) = ∫₀^(1/2) ln(1+2αx)/(1+4x²) dx, where I(0)=0 and I(1) is our target.</p><p><strong>Step 2:</strong> Differentiate under integral sign: dI/dα = ∫₀^(1/2) 2x/((1+2αx)(1+4x²)) dx</p><p><strong>Step 3:</strong> Partial fractions decomposition: 2x/((1+2αx)(1+4x²)) = A/(1+2αx) + (Bx+C)/(1+4x²). Solving: A = α/(1+α²), B = -α/(1+α²), C = 2/(1+α²)</p><p><strong>Step 4:</strong> Integrate each term from 0 to 1/2: The first term gives ln(1+α)/[2(1+α²)]. The second and third combine to give (1/4)arctan(1)·1/(1+α²) = π/16(1+α²)</p><p><strong>Step 5:</strong> Integrate dI/dα with respect to α from 0 to 1: I(1) = [ln(1+α)·arctan(α)/(1+α²)]₀¹ = (ln 2·π/4)/2 = <strong>π ln(2)/8</strong></p><p>∴ Answer: B</p>
Correct Answer: B