# Primitive of Function of Arccosine

## Theorem

$\displaystyle \int F \paren {\arccos \frac x a} \rd x = -a \int F \paren u \sin u \rd u$

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

## Proof

First note that:

 $\displaystyle u$ $=$ $\displaystyle \arccos \frac x a$ $\displaystyle \implies \ \$ $\displaystyle x$ $=$ $\displaystyle a \cos u$ Definition of Arccosine

Then:

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

$\blacksquare$