# Primitive of Arccosine of x over a over x

## Theorem

$\displaystyle \int \frac {\arccos \frac x a \rd x} x = \frac \pi 2 \ln \size x - \int \frac {\arcsin \frac x a \rd x} x + C$

## Proof

 $\displaystyle \int \frac {\arccos \frac x a \rd x} x$ $=$ $\displaystyle \int \frac {\paren {\frac \pi 2 - \arcsin \frac x a} \rd x} x$ Sum of Arcsine and Arccosine $\displaystyle$ $=$ $\displaystyle \frac \pi 2 \int \frac {\d x} x - \int \frac {\arcsin \frac x a \rd x} x$ Linear Combination of Integrals $\displaystyle$ $=$ $\displaystyle \frac \pi 2 \ln \size x - \int \frac {\arcsin \frac x a \rd x} x + C$ Primitive of Reciprocal

$\blacksquare$