Primitive of x squared over Root of x squared minus a squared/Logarithm Form

Theorem

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

Proof
With a view to expressing the problem 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: