Definite Integral from 0 to Half Pi of Logarithm of Cosine x

Theorem

 * $\displaystyle \int_0^{\pi/2} \map \ln {\cos x} \rd x = -\frac \pi 2 \ln 2$

Proof
By Definite Integral from 0 to $\dfrac \pi 2$ of $\map \ln {\sin x}$: Lemma we have:


 * $\displaystyle \int_0^{\pi/2} \map \ln {\sin x} \rd x = \int_0^{\pi/2} \map \ln {\cos x} \rd x$

and:


 * $\displaystyle \int_0^{\pi/2} \map \ln {\sin x} \rd x = -\frac \pi 2 \ln 2$

The result follows.