Primitive of Arccosine of x over a over x

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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

$\blacksquare$


Also see


Sources