Primitive of Reciprocal of a squared minus x squared

Theorem

 * $\displaystyle \int \frac {\mathrm d x} {a^2 - x^2} = \frac 1 a \tanh^{-1} {\frac x a} + C$

where $x^2 < a^2$.

Proof
Let:

Also see

 * Primitive of Reciprocal of Variable Squared minus Constant Squared for the case $x^2 > a^2$.