Definite Integration
Properties of Definite Integrals
Premium Question
Grade 12
Question:
$\int_0^1 \frac{\ln(1 + x)}{1 + x^2} \, dx$ equals:
(1) $\frac{\pi}{8} \ln 2$
(2) $\frac{\pi}{4} \ln 2$
(3) $\frac{\pi}{2} \ln 2$
(4) $\pi \ln 2$
Step-by-Step Solution
Key Concept: Substitute $x = \tan \theta$ so that $\frac{dx}{1 + x^2} = d\theta$, converting the integral into $\int_0^{\pi/4} \ln(1 + \tan \theta) \, d\theta$, and then apply the King's rule.
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Correct Answer: (1)