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

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Theorem

$\ds \int_0^{\pi/2} \paren {\map \ln {\cos x} }^2 \rd x = \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}$


Proof

\(\ds \int_0^{\pi/2} \paren {\map \ln {\cos x} }^2 \rd x\) \(=\) \(\ds \int_0^{\pi/2} \paren {\map \ln {\map \cos {\frac \pi 2 - x} } }^2 \rd x\)
\(\ds \) \(=\) \(\ds \int_0^{\pi/2} \paren {\map \ln {\sin x} }^2 \rd x\) Cosine of Complement equals Sine
\(\ds \) \(=\) \(\ds \frac \pi 2 \paren {\ln 2}^2 + \frac {\pi^3} {24}\) Definite Integral from $0$ to $\dfrac \pi 2$ of $\paren {\map \ln {\sin x} }^2$

$\blacksquare$


Sources