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

Theorem

$\ds \int \sqrt {a^2 - x^2} \rd x = \frac {x \sqrt {a^2 - x^2} } 2 - \frac {a^2} 2 \arccos \frac x a + C$

Let:

$\ds \int \sqrt {a^2 - x^2} \rd x$

$=$

$\ds \frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \frac x a + C$

$\ds $

$=$

$\ds \frac {x \sqrt {a^2 - x^2} } 2 + \paren {\dfrac \pi 2 - \frac {a^2} 2 \arcsin \frac x a} + C$

$\ds $

$=$

$\ds \frac {x \sqrt {a^2 - x^2} } 2 - \frac {a^2} 2 \arccos \frac x a + C$

subsuming $\dfrac \pi 2$ into arbitrary constant $C$

$\blacksquare$