Lời giải

Đặt \( t = \frac{\pi}{4} - x \Rightarrow x = \frac{\pi}{4} - t \), \( dx = -dt \)  

\( I = - \int_{\frac{\pi}{4}}^{0} \ln \left(1 + \tan \left(\frac{\pi}{4} - t\right)\right) dt \)

\(   = \int_{0}^{\frac{\pi}{4}} \ln \left(1 + \frac{1 - \tan t}{1 + \tan t}\right) \, dt \)

\( = \int_{0}^{\frac{\pi}{4}} \ln \left(\frac{2}{1 + \tan t}\right) \, dt \)

\( = \int_{0}^{\frac{\pi}{4}} \ln 2  - I \)

\(  \Rightarrow I = \frac{1}{2} \ln 2 \bigg|_0^{\frac{\pi}{4}} = \frac{\pi}{8} \ln 2 \)