Question

Find the \(L\left (\frac{d}{dt}(\frac{sin⁡t}{t})\right)\).

(a) s×cot^-1 s-1

(b) s×tan^-1 s-1

(c) s×cot⁡(s)-1

(d) s×tan⁡(s)-1

The question was asked during an interview for a job.

I'd like to ask this question from Table of General Properties of Laplace Transform in portion Laplace Transform of Engineering Mathematics

Answer 1

The correct choice is (a) s×cot^-1 s-1

To elaborate: In the given question,

L(sint)=\(\frac{1}{s^2+1}\)

By effect of division by t,

\(L(\frac{sin⁡t}{t})=\int_{s}^{\infty}\frac{1}{s^2+1} ds\)

=\(\frac{\pi}{2}-tan^{-1}s\)

=cot^-1s

By effect of derivative of Laplace Transform,

\(L\left (\frac{d}{dt}\left (\frac{sin⁡t}{t}\right )\right )=s×cot^{-1}s-1\).