# Primitive of Reciprocal of Root of x squared minus a squared/Logarithm Form/Proof 1

$\displaystyle \int \frac {\d x} {\sqrt {x^2 - a^2} } = \map \ln {x + \sqrt {x^2 - a^2} } + C$
 $\displaystyle \int \frac {\d x} {\sqrt {x^2 - a^2} }$ $=$ $\displaystyle \cosh^{-1} {\frac x a} + C'$ Primitive of Reciprocal of $\sqrt {x^2 - a^2}$: $\cosh^{-1}$ form $\displaystyle$ $=$ $\displaystyle \map \ln {\frac x a + \sqrt {\paren {\frac x a}^2 - 1} } + C'$ Definition of Real Inverse Hyperbolic Cosine $\displaystyle$ $=$ $\displaystyle \map \ln {\frac x a + \sqrt {\frac {x^2 - a^2} {a^2} } } + C'$ rearrangement $\displaystyle$ $=$ $\displaystyle \map \ln {\frac {x + \sqrt {x^2 - a^2} } a} + C'$ rearrangement $\displaystyle$ $=$ $\displaystyle \map \ln {x + \sqrt {x^2 - a^2} } - \ln a + C'$ Difference of Logarithms $\displaystyle$ $=$ $\displaystyle \map \ln {x + \sqrt {x^2 - a^2} } + C$ putting $C = -\ln a + C'$
$\blacksquare$