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

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Theorem

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


Proof

Let:

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


Also:

\(\ds x\) \(=\) \(\ds a \sin \theta\)
\(\ds \leadsto \ \ \) \(\ds a^2 - x^2\) \(=\) \(\ds a^2 - a^2 \sin^2 \theta\)
\(\ds \) \(=\) \(\ds a^2 \paren {1 - \sin^2 \theta}\)
\(\ds \) \(=\) \(\ds a^2 \cos^2 \theta\) Sum of Squares of Sine and Cosine
\(\text {(2)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \sqrt {a^2 - x^2}\) \(=\) \(\ds a \cos \theta\)


and:

\(\ds x\) \(=\) \(\ds a \sin \theta\)
\(\text {(3)}: \quad\) \(\ds \leadsto \ \ \) \(\ds \theta\) \(=\) \(\ds \map \arcsin {\frac x a}\) Definition of Real Arcsine


Thus:

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

$\blacksquare$


Also see


Sources

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