Primitive of Root of a squared minus x squared

From ProofWiki
Jump to navigation Jump to search

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:

\(\displaystyle x\) \(=\) \(\displaystyle a \sin \theta\)
\((1):\quad\) \(\displaystyle \implies \ \ \) \(\displaystyle \frac {\d x} {\d \theta}\) \(=\) \(\displaystyle a \cos \theta\) 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
\((2):\quad\) \(\displaystyle \implies \ \ \) \(\displaystyle \sqrt {a^2 - x^2}\) \(=\) \(\displaystyle a \cos \theta\)


and:

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

$\blacksquare$


Also see


Sources