If τ is time constant and ω is the applied frequency, a low pass RC filter acts as a pure integrator when

Low-pass filter is as shown:-

\( \frac{{{V_o}\left( s \right)}}{{{V_i}\left( s \right)}} = \frac{1}{{1 + sCR}} \)

\(= \frac{1}{{1 + s\tau }} = \frac{1}{{1 + j\omega \tau }}\)

\({V_o}\left( t \right) = {V_o}{e^{ - \frac{t}{\tau }}}\) .

For the capacitor to act as an integrator the capacitor should discharge slowly. And this discharging time depends upon the time constant (τ).

Less the time constant → faster will be the discharge

More the time constant → Slower will be the discharge

So, for the capacitor to act as an integrator,

ωτ ≫ 1