Definite Integral from 0 to Half Pi of Square of Logarithm of Cosine x
Jump to navigation
Jump to search
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
- 1968: Murray R. Spiegel: Mathematical Handbook of Formulas and Tables ... (previous) ... (next): $\S 15$: Definite Integrals involving Logarithmic Functions: $15.103$