[laTEX]x = tan t \Rightarrow I = \int_{0}^{\frac{\pi}{4}} ln (1 + tant) dt \\ \\ t = \frac{\pi}{4} - u \\ \\ I = \int_{0}^{\frac{\pi}{4}} ln (1 + tan( \frac{\pi}{4} - u)) dt \\ \\ I = \int_{0}^{\frac{\pi}{4}} ln (1 + \frac{1- tanx}{1+tanx}) dt \\ \\ I = \int_{0}^{\frac{\pi}{4}} ln (\frac{2}{1+tanx}) dt = I = \int_{0}^{\frac{\pi}{4}}ln 2.dx - I \\ \\ I = \frac{ln2}{2}x \big|_0^{\frac{\pi}{4}}[/laTEX]