Amplified Active Low Pass Filter with Capacitor in Place of Resistor
That is more a non-inverting integrator, if you analyze the input, it is a voltage divider, the voltage at the non inverting input is
$$V_p(s)=\frac{\frac{1}{sC_1}}{R_1+\frac{1}{sC_1}}V_1(s)$$
which simplified gives
$$V_p(s)=\frac{1}{1+sR_1C_1}V_1(s)   ...(1)$$
Now, R2 and C2 can be analyzed like a regular non-inverting amplifier
$$V_O(s)=\left(1+\frac{1}{sR_2C_2}\right)V_p(s)   $$
Rewriting
$$V_O(s)=\left(\frac{1+sR_2C_2}{sR_2C_2}\right)V_p(s)  ...(2) $$
If you substitute (1) in (2), you get
$$V_0(s)=\left(\frac{1+sR_2C_2}{sR_2C_2}\right)\left(\frac{1}{1+sR_1C_1}\right)V_1(s) ...(3)$$
Now here comes the interesting part, if \$C_1=C_2=C\$ and \$R_1=R_2=R\$
You get
$$V_O(s)=\frac{1}{sRC}V_i(s) $$
The inverse Laplace transform gives
$$v_o(t)=\frac{1}{RC}\int v_i(t)$$