Circuit analysis using an op-amp (non-ideal integrator)
Assuming your opamp is ideal, meaning that it can source or sink infinite current, draws no current at the inputs and has no offset voltage. You can neglect R3 and R4, since they don't have any effect to the output voltage.

Then you can just apply the formula for the inverting amplifier:

$$
V_{\mathrm{out}}=V_{\mathrm{in}}\frac{-Z_{\mathrm{f}}}{Z_{\mathrm{in}}}
$$
with Zf being the fedback impedance, and Zin the input impedance.
In your case:
$$Z_{\mathrm{f}}=R_2||\frac{1}{s C }=\frac{R_2}{R_2Cs+1}$$
Then:
$$
V_{\mathrm{out}}=V_{\mathrm{in}}\frac{\frac{R_2}{R_2Cs+1}}{R_1}=V_{\mathrm{in}}\frac{R_2}{R_1}\frac{1}{R_2Cs+1}
$$