Question

Consider a continuous-time signal \[ x(t) = -t^2 \left\{ u(t+4) - u(t-4) \right\} \] where \( u(t) \) is the continuous-time unit step function. Let \( \delta(t) \) be the continuous-time unit impulse function. The value of \[ \int_{-\infty}^{\infty} x(t)\delta(t+3) \, dt \] is:

• A. \( -9 \)
• B. \( 9 \)
• C. \( 3 \)
• D. \( -3 \)

Hint:

To evaluate \( \int x(t)\delta(t+a)\,dt \), apply the sifting property: it equals \( x(-a) \). Ensure that the value lies within the domain where \( x(t) \) is defined and non-zero.

Answer 1

The Correct Option is A

Given:
\[ x(t) = -t^2\{u(t+4) - u(t-4)\} \]

Step 1: Use the sifting property of delta function

\[ \int_{-\infty}^{\infty} x(t)\,\delta(t+3)\,dt = x(-3) \]

Step 2: Evaluate \(x(-3)\)

First check the unit step functions:

\[ u(-3+4) = u(1) = 1,\quad u(-3-4) = u(-7) = 0 \]

Hence,

\[ u(t+4) - u(t-4) = 1 \quad \text{at } t=-3 \]

Step 3: Substitute \(t=-3\)

\[ x(-3) = -(-3)^2 \times 1 = -9 \]

Final Answer:

\[ \boxed{-9} \]