Primitive of x squared over Root of x squared plus a squared/Inverse Hyperbolic Sine Form

Theorem

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

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

let:

and let:

Then: