Arcsine as Integral/Lemma 1

Lemma
Let $\sin_A$ be the analytic sine function for real numbers.

Let $\arcsin_A$ denote the real arcsine function.

Then:
 * $\ds \map {\arcsin_A} x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$

Proof
For this proof only, let $\sin_A$ be the analytic sine function.

Consider:
 * $\ds \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$

Let:
 * $x = \sin_A \theta \iff x = \map {\arcsin_A} \theta$

Then: