Primitive of Function of Arccosine

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

$\blacksquare$