Primitive of Root of a squared minus x squared/Arcsine Form

Theorem

 * $\displaystyle \int \sqrt {a^2 - x^2} \ \mathrm d x = \frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a + C$

Proof
Let:

Also:

and:

Thus:

Also see

 * Primitive of $\sqrt{x^2 + a^2}$
 * Primitive of $\sqrt{x^2 - a^2}$