Integrator and Differentiator

Integrator (capacitor in feedback):

$V_{out} = -\frac{1}{RC}\int V_{in}\,dt$

A square wave in gives a triangle wave out. Practical integrators add a large resistor across $C$ to stop the output drifting to the rail because of input offset.

Differentiator (capacitor at input):

$V_{out} = -RC\frac{dV_{in}}{dt}$

Differentiators amplify high-frequency noise, so a small series resistor and feedback capacitor are added to limit the bandwidth.