Limit^limit^...

Given the limit:

$\lim_{x \to 1} \frac {^nx\ -\ ^{n-1}x}{(1-x)^n} = \begin{cases} a & \text{if }n \text{ is even} \\ b & \text{if }n \text{ is odd} \end{cases}$

Compute

$\int_b^a \frac 1{\sqrt{1-x^2}} \ dx$

Notation: $^ka = \underbrace{a^{a^{a^{a^{\cdot^{\cdot^{\cdot^a}}}}}}}_{\text{Number of }a\text{'s}=k}$ denotes tetration .