Primitive of Function of Arcsine

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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

$\blacksquare$


Also see


Sources