An integral controller accumulates error over time:

$$u(t)=K_I\int_0^t e(\tau),d\tau.$$

Its transfer function is

$$C(s)=\frac{K_I}{s}.$$

As long as a nonzero error persists, the integral continues changing the control action. This lets integral feedback remove steady offsets that proportional control alone can leave behind.

The factor $1/s$ adds a pole at the origin to the loop transfer function and increases the system type. That improves low-frequency tracking and rejection of constant disturbances.

Integral action also adds dynamics. Too much integral gain can increase overshoot or create oscillation, and actuator saturation can cause the stored integral to grow excessively, a phenomenon called integral windup.

Integral control trades extra dynamic complexity for the ability to drive persistent error toward zero.