Question

Find the \(L\left (\frac{sinh⁡(at)}{t}\right )\).

(a) \(\frac{1}{2} log⁡ \left (\frac{s×a}{s-a}\right )\)

(b) \(\frac{1}{2} log⁡ \left (\frac{s-a}{s+a}\right )\)

(c) \(\frac{1}{2} log⁡ \left (\frac{s+a}{s-a}\right )\)

(d) \(\frac{1}{3} log⁡ \left (\frac{s+a}{s-a}\right )\)

The question was asked in quiz.

My query is from Table of General Properties of Laplace Transform topic in portion Laplace Transform of Engineering Mathematics

Answer 1

The correct choice is (c) \(\frac{1}{2} log⁡ \left (\frac{s+a}{s-a}\right )\)

To elaborate: In the given question,

L(sinh⁡(at))=\(\frac{a}{s^2-a^2}\)

By effect of division by t,

\(L\left (\frac{sinh⁡(at)}{t}\right )=\int_{s}^{\infty}\frac{a}{s^2-a^2} ds\)

=\(a×\frac{1}{2a}×log⁡ \left (\frac{s-a}{s+a}\right ) \) in limits s to ∞

=\(\frac{1}{2} log⁡(0)-\frac{1}{2} log⁡ \left (\frac{s-a}{s+a}\right )\)

=\(\frac{1}{2} log⁡ \left (\frac{s+a}{s-a}\right )\).