The Laplace transform of $$e^{at} cos ωt$$​

Correct option is A

​$$\text{The Laplace transform of } e^{at} \cos(\omega t) \text{ is given by:} \\[6pt]\mathcal{L}\{e^{at} \cos(\omega t)\} = \frac{s - a}{(s - a)^2 + \omega^2} \\[12pt]\textbf{Derivation:} \\[6pt]1. \ \textbf{Standard Laplace Transform of } \cos(\omega t): \\[4pt]\mathcal{L}\{\cos(\omega t)\} = \frac{s}{s^2 + \omega^2} \\[10pt]2. \ \textbf{Applying the Exponential Shift Theorem:} \\[4pt]\text{If } \mathcal{L}\{f(t)\} = F(s), \text{ then:} \\[4pt]\mathcal{L}\{e^{at} f(t)\} = F(s - a) \\[10pt]\text{Applying this to } \cos(\omega t): \\[4pt]\mathcal{L}\{e^{at} \cos(\omega t)\} = \frac{s - a}{(s - a)^2 + \omega^2}$$​​