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

Theorem

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

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:

By Primitive of Cotangent Function:


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

So:

We have:

giving:


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

hence the result.