Definite Integration
Evaluation of definite integrals
Grade 12
Question:
<p>Evaluate \(\int_0^1 \dfrac{\ln(x+1)}{x^2+1}\,dx\)</p>
<p>(a) \(\pi\log(2)/8\)</p>
<p>(b) \(\pi\log(2)/4\)</p>
<p>(c) \(\pi\log(2)/2\)</p>
<p>(d) None of these</p>
Step-by-Step Solution
Key Concept: Use Feynman's parameter differentiation technique: introduce a parameter in the logarithm, differentiate with respect to it, evaluate the resulting integral, then integrate back. This converts the awkward ln(x+1) into a rational function that can be handled via arctangent integrals.
<p><strong>Step 1:</strong> Introduce parameter. Let I(a) = ∫₀¹ ln(ax+1)/(x²+1) dx, where I(0) = 0 and we seek I(1).</p><p><strong>Step 2:</strong> Differentiate with respect to a: dI/da = ∫₀¹ x/[(ax+1)(x²+1)] dx</p><p><strong>Step 3:</strong> Use partial fractions on x/[(ax+1)(x²+1)]. After decomposition and integration:</p><p>dI/da = [1/(1+a²)] · [a·ln(a+1) + π/2 - π/4 - (1/2)ln(1+a²)]</p><p><strong>Step 4:</strong> Integrate dI/da from 0 to 1:</p><p>I(1) = ∫₀¹ [a·ln(a+1)/(1+a²)] da + (π/8)ln(2) - (1/4)∫₀¹ ln(1+a²)/(1+a²) da</p><p><strong>Step 5:</strong> Evaluate the remaining integrals (using arctangent substitution u = arctan(a)):</p><p>∴ Answer: <strong>π·ln(2)/8</strong></p>
Correct Answer: A