Primitive of Logarithm of x squared minus a squared

Theorem

 * $\displaystyle \int \ln \left({x^2 - a^2}\right) \rd x = x \ln \left({x^2 - a^2}\right) - 2 x + a \ln \left({\frac {x + a} {x - a} }\right) + C$

for $x^2 > a^2$.

Proof
With a view to expressing the primitive in the form:
 * $\displaystyle \int u \frac {\rd v}{\rd x} \rd x = u v - \int v \frac {\rd u}{\rd x} \rd x$

let:

and let:

Then:

Also see

 * Primitive of $\ln \left({x^2 + a^2}\right)$