Hi,
Two issue:
Your negative Opamp supply is GND. And your noninverting input is GND, too.
--> you need an Opamp that can handle this.
The integrator is inverting. So with positive input, the output is travelling towards the negative supply (GND).
So the Opamp output will saturate near GND, now way for it to go negative.
Therefore one needs to assume that at t=0 the output of the Opamp is well in the positve area.
*******
Now that +In of the Opamp is GND .... then -In is GND, too. (regulated by the Opamp)
This means the complete integrator_input_voltage will be across the resistor.
Therefore you may easily calculate integrator_input_current.
Now that the current at the Opamp_-In is considered to be zero....al the current must flow through the capacitor.
C = I × t / V ----> with a constant input voltage you get a constant input current and this causes a constant output rise/fall_rate in V/s.
Therefore rearrange the capacitor formula this way:
V / t = I / C ....where I = V_input /R, the result is in V/s
V / t = V / ( C × R )
Let's assume you have 3V integrator_input_voltage. Then..
V / t = 3V / (1uF × 47k) = 63.83V/s ... or 63.83mV/ ms
The output voltage of an integrateor at t = 0 is undefined, but let's assume it is 4.0V
Then :
t = 0 --> Vout = 4.0V
t = 1ms --> Vout = 4.0V - 63.83 mV = 3.936 V
t = 2ms --> Vout = 3.936V - 63.83mV = 3.872V
...and so on.. until it saturates.
With zero input voltage .... the rise rate is zero, means a flat horizontal line. The voltage freezes.
*****
Now you asked about sine wave input.
It is similarely simple, but I use another approach.
For an inverting amplifier:
V_out = V_in × R_fb / R_in
R_fb = feedback resistor. In your case the capacitor with it's Xc.
So V_out = V_in × Xc / R_in.
In your case with 3Vpp, 100Hz sine wave input:
Xc at 100Hz = 1 / ( 2 × Pi × 100Hz × 1uF) = 1592 Ohms.
V_out = 3Vpp x 1592 Ohms / 47k = 0.102Vpp
Klaus