Primitive of Arcsecant of x over a/Formulation 2

Theorem

 * $\displaystyle \int \arcsec \frac x a \rd x = x \arcsec \frac x a - a \ln \size {x + \sqrt {x^2 - a^2} } + C$

for $x^2 > 1$.

$\displaystyle \arcsec \frac x a$ is undefined on the real numbers for $x^2 < 1$.

Proof
With a view to expressing the primitive 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:

We then have:

Let $x > 1$.

Then:

Similarly, let $x < -1$.

Then: