In the width of $${r_2} - {r_1}$$ total n turns presents.

$$\therefore$$ In 1 unit width $${n \over {{r_2} - {r_1}}}$$ turns presents.

$$\therefore$$ In the width of dr number of turns,

$$n' = {n \over {{r_2} - {r_1}}} \times dr$$

Magnetic field (dB) due to element of dr length is

$$dB = {{{\mu _0} \times I \times n'} \over {2r}}$$

$$ = {{{\mu _0}I} \over {2r}} \times {n \over {({r_2} - {r_1})}}$$

$$\therefore$$ Total magnetic field due to entire coil is,

$$\int {dB = \int_{{r_1}}^{{r_2}} {{{{\mu _0}I} \over {2r}} \times {n \over {({r_2} - {r_1})}}dr} } $$

$$ \Rightarrow B = {{{\mu _0}In} \over {2({r_2} - {r_1})}}\int_{{r_1}}^{{r_2}} {{{dr} \over r}} $$

$$ = {{{\mu _0}In} \over {2({r_2} - {r_1})}} \times \left[ {\log _e^r} \right]_{{r_2}}^{{r_1}}$$

$$ = {{{\mu _0}In} \over {2({r_2} - {r_1})}} \times \left( {\log _e^{{r_2}} - \log _e^{{r_1}}} \right)$$

$$ = {{{\mu _0}In} \over {2({r_2} - {r_1})}} \times {\log _e}{{{r_2}} \over {{r_1}}}$$