1. Evaluate the following integral:

2. Draw the complex spectrum (both magnitude and phase) for the following signals:

(a) x(t) = 10 cos(200πt) cos(2000πt + π/2)

(b) x(t) = 1 + 10 cos(120πt) + 5 sin(300πt)

3. Compute the convolution between the following two signals:

2. Determine the Laplace transform of the following signals:

(a) cos(3t)U(t)

(b) e^-10t U(t)

(c) e^-10t cos(3t)U(t)

(d) e^-10t cos(3t - 1)U(t)

3. A continuous-time signal x(t) has the Laplace Transform:

Determine the Laplace Transform V(s) of the following:

(a) v(t) = x(3t - 4)U(3t - 4)

(b) v(t) = tx(t)

(c) v(t) = d²x(t)/dt²

4. Compute the Laplace Transform for each of the following:

5. A Linear time-invariant continuous-time system has the impulse response:

h(t) = e^-t + sin t, t is greater or equal to 0

(a) Compute the step response g(t) for all t greater or equal to 0.

(b) Compute the output response y(t) for all t greater or equal to 0 when the input is:

x(t) = U(t) - U(t-2) , with no initial energy

6. Determine he final values of each of the signals whose Laplace transforms are given below. If there is no final value, state why not.

(a) X(s) = 4 / (s² + s)

(b) X(s = (3s + 4) / (s² + s)