Definite Integral from 0 to Half Pi of Logarithm of Sine x by Cosine of 2nx
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Theorem
For $n \in \N_{>0}$:
- $\ds \int_0^{\pi/2} \map \ln {\sin x} \cos 2 n x \rd x = -\frac \pi {4 n}$
Proof
First we have:
| \(\ds \lim_{x \mathop \to 0} \map \ln {\sin x} \sin 2 n x\) | \(=\) | \(\ds \lim_{x \mathop \to 0} \frac {\map \ln {\sin x} } {\csc 2 n x}\) | Definition of Cosecant | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{x \mathop \to 0} \frac {\cot x} {- 2 n \cot 2 n x \csc 2 n x}\) | L'Hôpital's Rule: Corollary $2$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \lim_{x \mathop \to 0} \frac {\cos x} {- 2 n \cos 2 n x} \frac {\sin^2 2 n x} {\sin x}\) | Definition of Cosecant, Definition of Cotangent | |||||||||||
| \(\ds \lim_{x \mathop \to 0} \frac {\sin^2 2 n x} {\sin x}\) | \(=\) | \(\ds \lim_{x \mathop \to 0} \frac {4 n \sin 2 n x \cos 2 n x} {\cos x}\) | L'Hôpital's Rule | |||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) |
hence $\ds \lim_{x \mathop \to 0} \map \ln {\sin x} \sin 2 n x = 0$.
Thus:
| \(\ds \int_0^{\pi/2} \map \ln {\sin x} \cos 2 n x \rd x\) | \(=\) | \(\ds \frac 1 {2 n} \int_0^{\pi/2} \map \ln {\sin x} \map \rd {\sin 2 n x}\) | Primitive of $\cos a x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds \frac 1 {2 n} \paren {\bigintlimits {\map \ln {\sin x} \sin 2 n x} 0 {\pi/2} - \int_0^{\pi/2} \sin n x \map \rd {\map \ln {\sin x} } }\) | Integration by Parts | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac 1 {2 n} \int_0^{\pi/2} \sin n x \map \rd {\map \ln {\sin x} }\) | From above | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac 1 {2 n} \int_0^{\pi/2} \sin 2 n x \frac {\cos x} {\sin x} \rd x\) | Primitive of Cotangent Function | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac 1 {2 n} \int_0^{\pi/2} \frac {\sin \paren {2 n + 1} x + \sin \paren {2 n - 1} x} {2 \sin x} \rd x\) | Werner Formula for Sine by Cosine | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac 1 {4 n} \int_0^{\pi} \frac {\sin \paren {\paren {2 n + 1} u/2} + \sin \paren {\paren {2 n - 1} u/2} } {2 \sin \paren {u/2} } \rd u\) | substituting $u = 2 x$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac 1 {4 n} \int_0^{\pi} \paren {\frac 1 2 + \sum_{k \mathop = 1}^n \map \cos {k u} + \frac 1 2 + \sum_{k \mathop = 1}^{n - 1} \map \cos {k u} } \rd u\) | Lagrange's Cosine Identity | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac 1 {4 n} \int_0^{\pi} 1 \rd u\) | All integrals involving $\cos k u$ evaluate to $0$ | |||||||||||
| \(\ds \) | \(=\) | \(\ds -\frac \pi {4 n}\) | Definite Integral of Constant |
$\blacksquare$