Primitive of Reciprocal of x by Root of x squared minus a squared

Theorem

Arccosine Form

$\ds \int \frac {\d x} {x \sqrt {x^2 - a^2} } = -\frac 1 a \arcsin \size {\frac a x} + C$

for $0 < a < \size x$.

Arcsecant Form

$\ds \int \frac {\d x} {x \sqrt {x^2 - a^2} } = \frac 1 a \arcsec \size {\frac x a} + C$

for $0 < a < \size x$.

Arcsine Form

$\ds \int \frac {\d x} {x \sqrt {x^2 - a^2} } = -\frac 1 a \arcsin \size {\frac a x} + C$

for $0 < a < \size x$.