Time-domain integration insight: What is the result of double integrating the unit step function u(t)?

Introduction / Context:
Understanding how basic signals transform under integration is essential in control and signal processing. The unit step u(t) integrates to a ramp, and integrating again produces a quadratic-in-time function—a parabola—scaled and gated by u(t).

Given Data / Assumptions:

• Unit step u(t) = 1 for t ≥ 0, 0 for t < 0.
• We consider causal integration with zero initial conditions.
• We seek the functional form after two integrations.

Concept / Approach:

First integral of u(t) yields the ramp r(t) = t u(t). The second integral integrates t from 0 to t, generating a quadratic term. Multiplying by u(t) ensures causality (zero for t < 0).

Step-by-Step Solution:

Verification / Alternative check:

Differentiating (t^2/2) u(t) twice recovers u(t) (ignoring distributional impulses at t = 0 that depend on initial condition conventions), confirming correctness for standard engineering treatment.

Why Other Options Are Wrong:

• Impulse or doublet: these arise from differentiation of step or ramp, not integration.
• Ramp: that is the result of a single integration, not double.
• Constant: integrating a constant once gives a ramp, twice gives a parabola, not a constant.

Common Pitfalls:

• Confusing integration with differentiation outcomes (e.g., thinking step → impulse).
• Forgetting the u(t) gating in the final expression.

Final Answer:

A parabola