Primitive of x squared over Root of x squared minus a squared/Inverse Hyperbolic Cosine Form

Theorem

 * $\displaystyle \int \frac {x^2 \rd x} {\sqrt {x^2 - a^2} } = \frac {x \sqrt {x^2 - a^2} } 2 + \frac {a^2} 2 \cosh^{-1} \frac x a + C$

for $x > a$.

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

let:

and let:

Then:

Note that because:
 * $\cosh^{-1} \frac x a$ is defined for $x \ge a$ only

and:
 * $\dfrac {x^2} {\sqrt {x^2 - a^2} } is not defined for $x = a$

$x$ is constrained as indicated.

Also see

 * Primitive of Reciprocal of $\dfrac {x^2} {\sqrt {x^2 + a^2} }$
 * Primitive of Reciprocal of $\dfrac {x^2} {\sqrt {a^2 - x^2} }$