# Primitive of Function of Arcsine

## Theorem

$\displaystyle \int F \left({\arcsin \frac x a}\right) \ \mathrm d x = a \int F \left({u}\right) \cos u \ \mathrm d u$

where $u = \arcsin \dfrac x a$.

## Proof

First note that:

 $\displaystyle u$ $=$ $\displaystyle \arcsin \frac x a$ $\displaystyle \implies \ \$ $\displaystyle x$ $=$ $\displaystyle a \sin u$ Definition of Arcsine

Then:

 $\displaystyle u$ $=$ $\displaystyle \arcsin \frac x a$ $\displaystyle \implies \ \$ $\displaystyle \frac{\mathrm d u} {\mathrm d x}$ $=$ $\displaystyle \frac 1 {\sqrt {a^2 - x^2} }$ Derivative of Arcsine Function: Corollary $\displaystyle \implies \ \$ $\displaystyle \int F \left({\arcsin \frac x a}\right) \ \mathrm d x$ $=$ $\displaystyle \int F \left({u}\right) \ \sqrt {a^2 - x^2} \ \mathrm d u$ Primitive of Composite Function $\displaystyle$ $=$ $\displaystyle \int F \left({u}\right) \ \sqrt {a^2 - a^2 \sin^2 u} \ \mathrm d u$ Definition of $x$ $\displaystyle$ $=$ $\displaystyle \int F \left({u}\right) \ a \sqrt {1 - \sin^2 u} \ \mathrm d u$ $\displaystyle$ $=$ $\displaystyle \int F \left({u}\right) \ a \cos u \ \mathrm d u$ Sum of Squares of Sine and Cosine $\displaystyle$ $=$ $\displaystyle a \int F \left({u}\right) \ \cos u \ \mathrm d u$ Primitive of Constant Multiple of Function

$\blacksquare$