Question

The Laplace transform of the signal \(\int_0^t e^{-3t}\)  cos⁡2t dt is _____________

(a) \(\frac{-(s+3)}{s[(s+3)^2+4]}\)

(b) \(\frac{(s+3)}{s[(s+3)^2+4]}\)

(c) \(\frac{s(s+3)}{(s+3)^2+4}\)

(d) \(\frac{-s(s+3)}{(s+3)^2+4}\)

This question was addressed to me in an interview for job.

This intriguing question originated from Common Laplace Transforms topic in portion Laplace Transform and System Design of Signals and Systems

Answer 1

Right option is (b) \(\frac{(s+3)}{s[(s+3)^2+4]}\)

For explanation: Laplace transform of e-at cost u (t) = \(\frac{s+a}{(s+a)^2+1}\)

Let p (t) = e^-3t cos2t u (t)

Laplace transform of p (t), p(s) = \(\frac{s+3}{(s+3)^2+4}\)

∴ Laplace transform of \(\int_{-∞}^t \,p(t)dt  = \frac{1}{s} \int_{-∞}^0 \,p(t)dt + \frac{p(s)}{s}\)

Or, X(s) = \(\frac{(s+3)}{s[(s+3)^2+4]}\).