Evaluate : \[\int_{0}^{\pi / 4} \log (1+\tan \mathrm{x}) d \mathrm{x} \]

Let $\displaystyle \mathrm{I}=\int_{0}^{\pi / 4} \log (1+\tan \mathrm{x}) \mathrm{dx}$\begin{aligned} & =\int_{0}^{\pi / 4} \log \left(1+\tan \left(\frac{\pi}{4}-\mathrm{x}\right)\right) d \mathrm{x} \quad\left[\text { Using } \int_{0}^{a} f(\mathrm{x}) d \mathrm{x}=\int_{0}^{a} f(a-\mathrm{x}) d \mathrm{x}\right] & =\int_{0}^{\pi / 4} \log \left(1+\frac{1-\tan \mathrm{x}}{1+\tan \mathrm{x}}\right) \mathrm{dx} & =\int_{0}^{\pi / 4} \log \frac{2}{1+\tan \mathrm{x}} \mathrm{dx} & 2 \mathrm{I}=\int_{0}^{\pi / 4} \log 2 d \mathrm{x}=\frac{\pi}{4} \log 2 & \mathrm{I}=\frac{\pi}{8} \log 2 \end{aligned} $$