Primitive of Function of Root of a squared minus x squared

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Theorem

$\ds \int \map F {\sqrt {a^2 - x^2} } \rd x = a \int \map F {a \cos u} \cos u \rd u$

where $x = a \sin u$.


Proof

First note that:

\(\ds x\) \(=\) \(\ds a \sin u\)
\(\ds \leadsto \ \ \) \(\ds \sqrt {a^2 - x^2}\) \(=\) \(\ds \sqrt {a^2 - \paren {a \sin u}^2}\)
\(\ds \) \(=\) \(\ds a \sqrt {1 - \sin^2 u}\) taking $a$ outside the square root
\(\text {(1)}: \quad\) \(\ds \) \(=\) \(\ds a \cos u\) Sum of Squares of Sine and Cosine


Then:

\(\ds x\) \(=\) \(\ds a \sin u\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d x} {\d u}\) \(=\) \(\ds a \cos u\) Derivative of Sine Function
\(\ds \leadsto \ \ \) \(\ds \int \map F {\sqrt {a^2 - x^2} } \rd x\) \(=\) \(\ds \int a \map F {\sqrt {a^2 - x^2} } \cos u \rd u\) Integration by Substitution
\(\ds \) \(=\) \(\ds \int a \map F {a \cos u} \cos u \rd u\) from $(1)$
\(\ds \) \(=\) \(\ds a \int \map F {a \cos u} \cos u \rd u\) Primitive of Constant Multiple of Function

$\blacksquare$


Sources