试证：∫(π／3,π／6)ln(tanx)dx=0

elim

题：试证 \(\displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\ln(\tan{x})dx=0\)

zeta2

\[\begin{aligned}
&\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\ln (\tan x)\mathrm{d}x\\
=&\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\ln(\tan x)\mathrm{d}x+\underbrace{\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}\ln(\tan x)\mathrm{d}x}_{x\mapsto \frac{\pi}{2}-x}\\
=&\int_{\frac{\pi}{6}}^{\frac{\pi}{4}}\ln(\tan x)\mathrm{d}x+\int_{\frac{\pi}{4}}^{\frac{\pi}{6}}\ln(\tan (\frac{\pi}{2}-x))\mathrm{d}(\frac{\pi}{2}-x)\\
=&0
\end{aligned}\]

luyuanhong

楼上 zeta2 的解答很好！已收藏。