Primitive of x squared over x squared minus a squared/Inverse Hyperbolic Cotangent Form

Theorem

$\ds \int \frac {x^2 \rd x} {x^2 - a^2} = x - a \coth^{-1} \frac x a + C$

for $x^2 > a^2$.

Let:

$\ds \int \frac {x^2 \rd x} {x^2 - a^2}$

$=$

$\ds \int \frac {x^2 - a^2 + a^2} {x^2 - a^2} \rd x$

$\ds $

$=$

$\ds \int \frac {x^2 - a^2} {x^2 - a^2} \rd x + \int \frac {a^2} {x^2 - a^2} \rd x$

$\ds $

$=$

$\ds \int \d x + a^2 \int \frac {\d x} {x^2 - a^2}$

$\ds $

$=$

$\ds x + a^2 \int \frac {\d x} {x^2 - a^2} + C$

$\ds $

$=$

$\ds x + a^2 \paren {-\frac 1 a \coth^{-1} {\frac x a} } + C$

$\ds $

$=$

$\ds x - a \coth^{-1} {\frac x a} + C$

simplifying

$\blacksquare$