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

Theorem

Inverse Hyperbolic Secant Form

$\displaystyle \int \frac {\mathrm d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \operatorname{sech}^{-1} {\frac x a} + C$

Logarithm Form

$\displaystyle \int \frac {\d x} {x \sqrt {a^2 - x^2} } = -\frac 1 a \map \ln {\frac {a + \sqrt {a^2 - x^2} } x} + C$