Primitive of Reciprocal of x by Root of x squared minus a squared/Arcsine Form

Theorem

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

for $0 < a < \size x$.

Proof
We have that $\sqrt {x^2 - a^2}$ is defined only when $x^2 > a^2$, that is, either:
 * $x > a$

or:
 * $x < -a$

where it is assumed that $a > 0$.

Hence:

Also see

 * Primitive of Reciprocal of $x \sqrt {x^2 + a^2}$
 * Primitive of Reciprocal of $x \sqrt {a^2 - x^2}$