Calcul de \(~\int_0^{\pi/2} \ln(\cos x)\,dx\)

Soient: \[I = \displaystyle \int_0^{\pi/2} \ln(\cos x)\,dx \qquad \text{et}\qquad J = \displaystyle \int_0^{\pi/2} \ln(\sin x)\,dx\].

  1. Montrer que \(I\) et \(J\) existent et que \(I=J\).
  2. En considérant \(I+J\), calculer \(I\).