Let's collide and then integrate!

Classical Mechanics Level 4

Three identical discs $$A,B$$ and $$C$$ are initially kept at rest on a smooth horizontal plane. The disc $$A$$ is set in motion with velocity $$v$$ after which it experiences an elastic collision simultaneously with both the discs $$B$$ and $$C$$. The distance between the centers of discs $$B$$ and $$C$$ prior to the collision is $$\eta$$ times greater than the diameter of each disc. If the range of values for $$\eta$$ for which the disc $$A$$, after collision recoils back with some velocity can be expressed as $$\eta \in [a, \sqrt{b})$$, where $$a$$ and $$b$$ are coprime positive integers and $$b$$ is square-free, then evaluate the value of

$\large \displaystyle \int_0^1 \dfrac{x^b - x^a}{\ln x} \,dx .$