Integral Time and KI in PID
Various forms of the PID equation are possible, but from your description it appears you have: \$(K_p + 1/T_i s + K_ds)\$. Generally, \$K_i = 1/T_i\$.

Other common forms are: \$(K_p + K_i/s + K_ds)\$ ; \$K_p(1 + 1/T_i s + T_d s)\$

Arguably, the most useful form is: \$ \frac{K_p (1 + T_i s + T_i T_d s^2)} {T_i s} \$ as this is the easiest to build into a transfer function, and also the proportional gain operates on all three terms and is identifiable as a 'loop gain'. This form is also very useful in root locus analysis as it clearly identifies the PID's open loop pole, at \$s=0\$, and two open loop zeros, with the integral and derivative times, \$T_i\$ and \$T_d\$, allowing the zeros to be placed at appropriate positions in the s-plane.