# Arcsine as Integral

## Theorem

$\ds \map \arcsin x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$

## Proof

### Lemma 1

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

Let $\arcsin_A$ denote the real arcsine function.

Then:

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

$\Box$

### Lemma 2

Let $\sin_G$ be the geometric sine.

$\arcsin_G$ is the inverse of this function.

$\ds \map {\arcsin_G} x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$

$\Box$

$\ds \map \arcsin x = \map {\arcsin_A} x = \map {\arcsin_G} x = \int_0^x \frac {\d x} {\sqrt {1 - x^2} }$

$\blacksquare$