Integration Techniques and forms Integrate; $$\int \frac{1}{\sqrt{x^2 - a^2}} \, dx $$
I can get the answer by substituting $\sec t = \dfrac{x}{a}$ but I am left with $\ln\left|\dfrac{x + \sqrt{x^2 + a^2}}{a}\right|$. Granted, this is right but how would I get there without having the 1/a in the log?
Also is there a way to find $\int \sec t \space dt$ without having to multiply by $\dfrac{\sec t + \tan t}{\sec t + \tan t}$?
and finally, more of a general question; are there and integrals in which only a hyperbolic substitution can be used to integrate?

Regarding $\int\sec x\,dx$, you can do a tangent half-angle substitution, that is:
$t=\tan \frac{x}{2}$ and thus $\cos x=\frac{1-t^2}{1+t^2}$.
