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

Theorem

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

Proof
Let:

Also:

and:

Thus: