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Laplace transform of $$t \cos (at)$$ is
$$\frac{s^2 + a^2}{(s^2 - a^2)^2}$$
$$\frac{s}{(s^2 - a^2)^2}$$
$$\frac{s^2 - a^2}{(s^2 + a^2)^2}$$
$$\frac{s}{(s^2 + a^2)^2}$$
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