Arcsine as Integral


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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.

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

$\Box$


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

$\blacksquare$


Also see

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