Primitive of Root of x squared minus a squared over x squared/Inverse Hyperbolic Cosine Form

Theorem

$\ds \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x = \arcosh \dfrac x a - \frac {\sqrt {x^2 - a^2} } x + C$

for $x^2 \ge a^2$.

Let:

$\ds \int \frac {\sqrt {x^2 - a^2} } {x^2} \rd x$

$=$

$\ds \frac {-\sqrt {x^2 - a^2} } x + \ln \size {x + \sqrt {x^2 - a^2} } + C$

$\ds $

$=$

$\ds \frac {-\sqrt {x^2 - a^2} } x + \map \ln {x + \sqrt {x^2 - a^2} } + C$

as $x + \sqrt {x^2 - a^2} \ge 0$ for $x^2 \ge a^2$

$\ds $

$=$

$\ds \map \ln {x + \sqrt {x^2 - a^2} } - \ln a - \frac {\sqrt {x^2 - a^2} } x + C$

subsuming $\ln a$ into constant of integration and rearranging

$\ds $

$=$

$\ds \arcosh \dfrac x a - \frac {\sqrt {x^2 - a^2} } x + C$

$\blacksquare$