Primitive of Arcsecant of x over a/Formulation 2

Theorem

 * $\displaystyle \int \operatorname{arcsec} \frac x a \ \mathrm d x = x \operatorname{arcsec} \frac x a - a \ln \left\vert{x + \sqrt {x^2 - a^2} }\right\vert + C$

for $x^2 > 1$.

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

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

let:

and let:

We then have:

Let $x > 1$.

Then:

Similarly, let $x < -1$.

Then: