Definite Integration
Properties of Definite Integrals
Premium Question
Grade 12
Question:
$\int_0^{\pi/4} \ln(1 + \tan x) \, dx$ equals:
(1) $\frac{\pi}{8} \ln 2$
(2) $\frac{\pi}{4} \ln 2$
(3) $\frac{\pi}{2} \ln 2$
(4) $\frac{\pi}{8} \ln \frac{1}{2}$
Step-by-Step Solution
Key Concept: Use $\int_0^a f(x) \, dx = \int_0^a f(a - x) \, dx$ with $a = \pi/4$, and simplify $\tan(\frac{\pi}{4} - x) = \frac{1 - \tan x}{1 + \tan x}$ inside the resulting logarithm.
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Correct Answer: (1)