Confusion with the DC response of an RC integrator
The response at D.C. means the value of \$H(0)\$ where \$H(s)\$ is the frequency response. Imagine that you apply a fixed voltage to the integrator input. You will get a ramp as an output of the integrator ignoring the opamp supply rails. This ramp will have various harmonics, if you take its frequency response. Since it goes to infinity its D.C. component, which is the average value, is infinite too. Therefore there is a pole at \$s=0\$. This is a more qualitative way of explaining why \$H(s) = a / s\$.

To see why the ramp \$f(x) = a x\$ has an infinite response at D.C. consider the Laplace transform:
$$H(0) = \int_0^\infty a x \ \mathrm{d}x = \left.\frac{a x^2}{2}\right|_0^\infty = a \infty$$