I've been confused by this sort of problem and am not sure where to start. I know that for the integrator, v_out(t) = -1/RC∫v_in(t)dt and for the differentiator, v_out(t) = -RC*(dv_in/dt), but that's honestly all I've really got, and I'm not sure how that helps me in this case. I know I don't have much work here, but I really just don't know how to attack this and would appreciate a prod in the right direction.
Bill Bahn (a teacher) will probably respond shortly with excellent help, but in the meantime, I can tell you that your initial equations are incorrect. The 1st circuit is not a true integrator, and the second one is not a true differentiator.
Those equations are for ideal integrators/differentiators, which are what we've dealt with to this point. Is that not the case in this problem?
No, they are far from ideal. I am surprised that you have been given these problems without being taught the theory. I am not going to give you answers, but the theory is readily available.
Well, I don't know about "excellent", but we'll see where things go. As Ron H has already pointed out, the circuits are NOT ideal integrators or differentiators. This is easy to see by asking would would happen in the first case, the integrator, if the input is simply a constant, positive DC voltage. If it were an ideal integrator, would would the output waveform look like? Is this possible? Break the analysis into two pieces and three steps. 1) Find the relationship between the input voltage and the voltage at the non-inverting input of the opamp. 2) Find the relationship between the voltage at the non-inverting input of the opamp and the output voltage. 3) Combine the two. Make you best attempt at the first two steps and report back. If you can, do the third step as well, but if you aren't confident you got both of the first two then we can put that on hold for a bit.
I addition to the above comments I like to mention that there are no "ideal" integrators based on opamp RC circuits. All "integrators" are, in fact, first order low-pass circuits. However, for frequencies far above the lowpass 3dB corner frequency (factor 50..100) they can be regarded as "nearly ideal". For example, the classical inverting integrator (capacitive feedback) can be designed for corner frequencies of 1 Hz or even less. Thus, integration is possible above - let`s say - 100 Hz (up to a limit set by used opamp).