Question

Find the Laplace transform of (cos⁡2t)^3 u(t).

(a) \(\frac{s(s^2+28)}{(s^2+36)(s^2+4)}\)

(b) \(\frac{s(s^2+36)}{(s^2+28)(s^2+4)}\)

(c) \(\frac{s(s^2+4)}{(s^2+36)(s^2+28)}\)

(d) \(\frac{s}{(s^2+36)(s^2+4)}\)

The question was asked in an online interview.

My question is based upon The Laplace Transform topic in chapter Laplace Transform and System Design of Signals and Systems

Answer 1

The correct answer is (a) \(\frac{s(s^2+28)}{(s^2+36)(s^2+4)}\)

For explanation: Given  x(t)=(cos⁡2t)^3 u(t) = \(\frac{[cos⁡6t+3cos⁡2t]}{4}\) u(t)

X(s) = L{x(t)} = \(L[\frac{(cos⁡6t+3cos⁡2t)}{4} \,u(t)] = \frac{1}{4}\) {L[cos⁡6t u(t)]+3L[cos⁡2t u(t)]}

 = \(\frac{1}{4} \left(\frac{s}{s^2+(6)^2} + 3 \frac{s}{s^2+(2)^2}\right) = \frac{s(s^2+28)}{(s^2+36)(s^2+4)}\).