# Primitive of Root of a squared minus x squared

## Theorem

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

## Proof

Let:

 $\displaystyle x$ $=$ $\displaystyle a \sin \theta$ $\text {(1)}: \quad$ $\displaystyle \implies \ \$ $\displaystyle \frac {\d x} {\d \theta}$ $=$ $\displaystyle a \cos \theta$ Derivative of Sine Function

Also:

 $\displaystyle x$ $=$ $\displaystyle a \sin \theta$ $\displaystyle \implies \ \$ $\displaystyle a^2 - x^2$ $=$ $\displaystyle a^2 - a^2 \sin^2 \theta$ $\displaystyle$ $=$ $\displaystyle a^2 \left({1 - \sin^2 \theta}\right)$ $\displaystyle$ $=$ $\displaystyle a^2 \cos^2 \theta$ Sum of Squares of Sine and Cosine $\text {(2)}: \quad$ $\displaystyle \implies \ \$ $\displaystyle \sqrt {a^2 - x^2}$ $=$ $\displaystyle a \cos \theta$

and:

 $\displaystyle x$ $=$ $\displaystyle a \sin \theta$ $\text {(3)}: \quad$ $\displaystyle \implies \ \$ $\displaystyle \theta$ $=$ $\displaystyle \arcsin \left({\frac x a}\right)$ Definition of Real Arcsine

Thus:

 $\displaystyle \int \sqrt {a^2 - x^2} \rd x$ $=$ $\displaystyle \int \sqrt {a^2 - x^2} \, a \cos \theta \rd \theta$ Integration by Substitution from $(1)$ $\displaystyle$ $=$ $\displaystyle \int a^2 \cos^2 \theta \rd \theta$ substituting for $\sqrt {a^2 - x^2}$ from $(2)$ $\displaystyle$ $=$ $\displaystyle a^2 \int \cos^2 \theta \rd \theta$ Primitive of Constant Multiple of Function $\displaystyle$ $=$ $\displaystyle a^2 \frac {\sin \theta \cos \theta + \theta} 2 + C$ Primitive of Square of Cosine Function: Corollary $\displaystyle$ $=$ $\displaystyle \frac 1 2 a \sin \theta \, a \cos \theta + \frac {a^2 \theta} 2 + C$ rearranging $\displaystyle$ $=$ $\displaystyle \frac 1 2 x a \cos \theta + \frac {a^2 \theta} 2 + C$ substituting $x = a \sin \theta$ $\displaystyle$ $=$ $\displaystyle \frac 1 2 x \sqrt {a^2 - x^2} + \frac {a^2 \theta} 2 + C$ substituting $\sqrt {a^2 - x^2} = a \cos \theta$ from $(2)$ $\displaystyle$ $=$ $\displaystyle \frac {x \sqrt {a^2 - x^2} } 2 + \frac {a^2} 2 \arcsin \left({\frac x a}\right) + C$ substituting for $\theta = \arcsin \left({\dfrac x a}\right)$ from $(3)$

$\blacksquare$