Op-amp integrator basics — identify the feedback element used to realize ideal integration in the inverting configuration:

Introduction / Context:The classic inverting op-amp integrator replaces the feedback resistor with a capacitor, producing an output proportional to the time integral of the input. This building block appears in active filters, waveform generators, and control systems where precise slope control is needed.

Given Data / Assumptions:

• Ideal op-amp behavior with negative feedback.
• Input resistor R_in feeding the inverting input; noninverting input referenced properly.
• Capacitor C used in the feedback path; a large resistor in parallel with C may be added to limit DC gain.

Concept / Approach:The ideal transfer function is V_out(s) / V_in(s) = −1 / (R_in * C * s), the hallmark of integration. Physically, the inverting summing node current charges the feedback capacitor, causing the output to ramp until feedback balances KCL at the node. In practice, a bleed resistor across the capacitor stabilizes DC operating point and limits low-frequency gain.

Step-by-Step Solution:

Verification / Alternative check:Square-wave input produces a triangular output; a sine input shifts phase by −90° in the midband where the integrator behaves ideally.

Why Other Options Are Wrong:

Common Pitfalls:Omitting the bleed resistor and saturating at DC; exceeding op-amp slew-rate or bandwidth; neglecting input bias current compensation.

Final Answer:Correct — the feedback element is a capacitor