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

Theorem

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

Proof

By Definite Integral from $0$ to $\dfrac \pi 2$ of $\map \ln {\sin x}$: Proof 1 we have:

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

and:

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

The result follows.

$\blacksquare$

Sources

• 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Logarithmic Functions: $15.102$