allery$\displaystyle\int_0^{\pi/2} \log(\tan x)\, dx = ?$
A$-(\pi/2)\ln 2$
B$0$ (by King's symmetry between $\sin$ and $\cos$)
C$\pi\ln 2$
D$\ln 2$
Answer & Solution
Correct answer: B. $0$ (by King's symmetry between $\sin$ and $\cos$)
$I = \int \log\tan x = \int(\log\sin x - \log\cos x)$.
Both pieces equal by King's (previous problem), so the difference is 0.
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