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Show that if $ 0 \le f(t) \le Me^{at} $ for $ t \ge 0 $, where $ M $ and $ a $ are constants, then the Laplace transform $ F(s) $ exists for $ s > a $.

Hence $\int_{0}^{+\infty} M e^{(a-s) t} d t$ is convergent and then by the Comparison Theorem

we have that $\int_{0}^{+\infty} f(t) e^{-s t} d t$ is convergent and therefore the Laplace transform

$F(s)$ exists.

Integration Techniques

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