Primitive of Function of Root of a squared minus x squared

Theorem

 * $\displaystyle \int F \left({\sqrt {a^2 - x^2}}\right) \ \mathrm d x = a \int \cos u \ F \left({a \cos u}\right) \ \mathrm d u$

where $x = a \sin u$.

Proof
First note that:

Then: