Definite Integral from 0 to Half Pi of Logarithm of Sine x/Lemma
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Lemma for Definite Integral from 0 to Half Pi of Logarithm of Sine x
- $\ds \int_0^\pi \map \ln {\sin x} \rd x = 2 \int_0^{\pi/2} \map \ln {\sin x} \rd x$
Proof
We have:
| \(\ds \int_{\pi/2}^\pi \map \ln {\sin x} \rd x\) | \(=\) | \(\ds -\int_{\pi/2}^0 \map \ln {\map \sin {\pi - x} } \rd x\) | substituting $x \mapsto \pi - x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \int_0^{\pi/2} \map \ln {\sin x} \rd x\) | Reversal of Limits of Definite Integral, Sine of Supplementary Angle |
We can therefore write:
| \(\ds \int_0^\pi \map \ln {\sin x} \rd x\) | \(=\) | \(\ds \int_0^{\pi/2} \map \ln {\sin x} \rd x + \int_{\pi/2}^\pi \map \ln {\sin x} \rd x\) | Sum of Integrals on Adjacent Intervals for Integrable Functions | |||||||||||
| \(\ds \) | \(=\) | \(\ds 2 \int_0^{\pi/2} \map \ln {\sin x} \rd x\) |
$\blacksquare$