Definite Integration
Properties of Definite Integrals
IIT-JEE 2001
Grade 12
Question:
$\int_0^{\pi/2} \ln(\sin x) \, dx$ equals:
(1) $-\frac{\pi}{2} \ln 2$
(2) $\frac{\pi}{2} \ln 2$
(3) $-\pi \ln 2$
(4) $-\frac{\pi}{4} \ln 2$
Step-by-Step Solution
Key Concept: Use $\int_0^{\pi/2} f(x) \, dx = \int_0^{\pi/2} f(\frac{\pi}{2} - x) \, dx$ to relate $\ln(\sin x)$ to $\ln(\cos x)$, then add the resulting expressions and use $\sin x \cos x = \frac{1}{2} \sin 2x$.
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Correct Answer: (1)