Primitive of Reciprocal of Root of x squared minus a squared

Theorem

Inverse Hyperbolic Cosine Form

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

for $x^2 > a^2$.

Logarithm Form

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

Also see

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