Consider an elemental ring of radius $x$ and with thickness $dx$

Number of turns in the ring $ = dN = \frac{{Ndx}}{{b - a}}$

Magnetic field at the centre due to the ring element
$dB = \frac{{{\mu _0}(dN)i}}{{2x}} = \frac{{{\mu _0}i}}{2}.\frac{{Ndx}}{{(b - a)}}.\frac{1}{x}$

$\therefore $ Field at the centre
$ = \int_{}^{} {dB = \frac{{{\mu _0}Ni}}{{2(b - a)}}} \int_a^b {\frac{{dx}}{x}} $
$ = \frac{{{\mu _0}Ni}}{{2(b - a)}}\ln \frac{b}{a}.$