Definite Integral from 0 to 1 of Arcsine of x over x

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Theorem

$\ds \int_0^1 \frac {\arcsin x} x = \frac \pi 2 \ln 2$


Proof

Let:

$x = \sin \theta$

By Derivative of Sine Function, we have:

$\dfrac {\d x} {\d \theta} = \cos \theta$

We have, by Arcsine of Zero is Zero:

as $x \to 0$, $\theta \to \arcsin 0 = 0$.

By Arcsine of One is Half Pi, we have:

as $x \to 1$, $\theta \to \arcsin 1 = \dfrac \pi 2$.

We have:

\(\ds \int_0^1 \frac {\arcsin x} x \rd x\) \(=\) \(\ds \int_0^{\pi/2} \frac {\cos \theta \map \arcsin {\sin \theta} } {\sin \theta} \rd \theta\) substituting $x = \sin \theta$
\(\ds \) \(=\) \(\ds \int_0^{\pi/2} \theta \cot \theta \rd \theta\) Definition of Real Arcsine, Definition of Real Cotangent Function

By Primitive of Cotangent Function:

$\ds \int \cot \theta \rd \theta = \map \ln {\sin \theta} + C$

So:

\(\ds \int_0^{\pi/2} \theta \cot \theta \rd \theta\) \(=\) \(\ds \bigintlimits {\theta \map \ln {\sin \theta} } 0 {\frac \pi 2} - \int_0^{\pi/2} \map \ln {\sin \theta} \rd \theta\) Integration by Parts
\(\ds \) \(=\) \(\ds \frac \pi 2 \map \ln {\sin \frac \pi 2} - \lim_{\theta \to 0^+} \paren {\theta \map \ln {\sin \theta} } + \frac \pi 2 \ln 2\) Definite Integral from 0 to $\dfrac \pi 2$ of $\map \ln {\sin x}$
\(\ds \) \(=\) \(\ds -\lim_{\theta \to 0^+} \paren {\theta \map \ln {\sin \theta} } + \frac \pi 2 \ln 2\) Sine of Right Angle, Natural Logarithm of 1 is 0

We have:

\(\ds \lim_{\theta \to 0^+} \paren {\theta \map \ln {\sin \theta} }\) \(=\) \(\ds \lim_{\theta \to 0^+} \paren {\theta \map \ln {\frac {\sin \theta} \theta \theta} }\)
\(\ds \) \(=\) \(\ds \lim_{\theta \to 0^+} \paren {\theta \map \ln {\frac {\sin \theta} \theta} } + \lim_{\theta \to 0^+} \theta \ln \theta\) Sum of Logarithms, Sum Rule for Limits of Real Functions
\(\ds \) \(=\) \(\ds \paren {\lim_{\theta \to 0^+} \theta} \paren {\map \ln {\lim_{\theta \to 0^+} \frac {\sin \theta} \theta} } + \lim_{\theta \to 0^+} \theta \ln \theta\) Product Rule for Limits of Real Functions
\(\ds \) \(=\) \(\ds 0 \ln 1 + 0\) Limit of $\dfrac {\sin x} x$ at Zero, Limit of $x^n \paren {\ln x}^m$
\(\ds \) \(=\) \(\ds 0\)

giving:

$\ds \int_0^{\pi/2} \theta \cot \theta \rd \theta = \frac \pi 2 \ln 2$

hence the result.

$\blacksquare$


Sources

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