Question
Find the Laplace transform of
Answer :
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We are asked to find the Laplace transform of ( f(t) = \frac{\sin\sqrt{t}}{\sqrt{t}} ). Recall the Laplace transform definition: [ \mathcal{L}{f(t)} = \int_0^\infty e^{-st} f(t), dt ] Substitute ( f(t) = \frac{\sin \sqrt{t}}{\sqrt{t}} ): [ \mathcal{L}\left{\frac{\sin\sqrt{t}}{\sqrt{t}}\right} = \int_0^\infty e^{-st} \frac{\sin\sqrt{t}}{\sqrt{t}} , dt ] Use the substitution ( u = \sqrt{t} \implies t = u^2, , dt = 2u, du ): [ \int_0^\infty e^{-s
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