Primitive of Function of Arccosine

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Theorem

$\ds \int \map F {\arccos \frac x a} \rd x = -a \int \map F u \sin u \rd u$

where $u = \arccos \dfrac x a$.


Proof

First note that:

\(\ds u\) \(=\) \(\ds \arccos \frac x a\)
\(\ds \leadsto \ \ \) \(\ds x\) \(=\) \(\ds a \cos u\) Definition of Arccosine


Then:

\(\ds u\) \(=\) \(\ds \arccos \frac x a\)
\(\ds \leadsto \ \ \) \(\ds \frac {\d u} {\d x}\) \(=\) \(\ds \frac {-1} {\sqrt {a^2 - x^2} }\) Derivative of Arccosine Function: Corollary
\(\ds \leadsto \ \ \) \(\ds \int \map F {\arccos \frac x a} \rd x\) \(=\) \(\ds \int \map F u \paren {-\sqrt {a^2 - x^2} } \rd u\) Primitive of Composite Function
\(\ds \) \(=\) \(\ds \int \map F u \paren {-\sqrt {a^2 - a^2 \cos^2 u} } \rd u\) Definition of $x$
\(\ds \) \(=\) \(\ds \int \map F u a \paren {-\sqrt {1 - \cos^2 u} } \rd u\)
\(\ds \) \(=\) \(\ds \int \map F u a \paren {-\sin u} \rd u\) Sum of Squares of Sine and Cosine
\(\ds \) \(=\) \(\ds -a \int\map F u \sin u \rd u\) Primitive of Constant Multiple of Function

$\blacksquare$


Also see