Definite Integral from 0 to Quarter Pi of Logarithm of One plus Tan x

Theorem

 * $\displaystyle \int_0^{\pi/4} \map \ln {1 + \tan x} \rd x = \frac \pi 8 \ln 2$

Proof
Therefore:


 * $\displaystyle 2 \int_0^{\pi/4} \map \ln {1 + \tan x} \rd x = \frac \pi 4 \ln 2$

giving:


 * $\displaystyle \int_0^{\pi/4} \map \ln {1 + \tan x} \rd x = \frac \pi 8 \ln 2$